7
Correlation
7.1 Introduction
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This chapter deals with measuring dependence between points in a point pattern, using the concept
of correlation.
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Often the motivation for analysing point pattern data is to determine whether the points appear
to have been placed independently of each other, or whether they exhibit some kind of interpoint
dependence.
Figure 7.1 shows three archetypal point patterns representing ‘regularity’ (where points tend to
avoid each other), ‘independence’ (complete spatial randomness), and ‘clustering’ (where points
tend to be close together).
Figure 7.1. Classical trichotomy between regular (Left), independent (Middle), and clustered
(Right) point patterns. All three patterns are in the unit square.
A standard statistical tool for measuring dependence is correlation, or more generally covariance. In this chapter we explain how to define and measure covariance in a point process, in such a
way that (roughly speaking) the clustered point pattern in the right panel of Figure 7.1 has positive
covariance, the completely random pattern in the middle panel has zero covariance, and the regular
pattern in the left panel has negative covariance.
In statistical theory, correlation is classified as a second moment quantity. The ‘first moment’ of
a random variable X is its mean value; the ‘second moment’ or ‘mean square’ is the mean of X 2 .
Together the first and second moments of random variables determine important quantities such as
variance, standard deviation, covariance, and correlation.
Second moment quantities for point processes are intimately related to counting pairs of points,
or adding up contributions from each pair of points in the process. In a point process X, the squared
point count n(X ∩ B)2 can be interpreted as the number of pairs of points xi , x j in X which fall in
the nominated set B, including identical pairs where i = j. The second moment of n(X ∩ B) is the
expected number of pairs of points falling in B.
Correlation has the great virtue of being easy to calculate and handle, and is a powerful tool
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7.2 Manual methods
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for the data analyst. There are two important caveats. First, accurate measurement of correlation
requires faithful estimation of the mean (first moment) if we are to avoid problems of spurious correlation and confounding. In the point process context, this means that we must have good knowledge
of the intensity before we can trust the correlation. Second, the correlation is merely a summary
index of statistical association, not a characterisation of dependence or causality: ‘correlation is not
causation’. Using only correlations, we cannot discriminate between different possible causes of
spatial clustering.
We start by describing two simple, manual techniques which motivate and give insight into the more
advanced methods for assessing corrlation in point pattern data.
7.2.1
Greig-Smith plot and Morisita index
If the observation window is a rectangle, a simple strategy for assessing spatial correlation is to
subdivide the window into rectangular quadrats of equal size, and to count how often a pair of data
points falls in the same quadrat. If there are n data points altogether, there are n(n − 1) ordered pairs
of distinct points. If we use m quadrats and these are found to contain n1 , . . . , nm points, respectively,
then the jth quadrat contains n j (n j − 1) ordered pairs of distinct points. The total number of ordered
pairs of distinct points which fall inside the same quadrat is thus ∑ j n j (n j − 1). The ratio
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∑ j n j (n j − 1)
n(n − 1)
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is the fraction of all pairs of data points in which both points fall in the same quadrat. In a completely
random (homogeneous Poisson) process, where points are independent of each other, two points fall
in the same quadrat with probability 1/m, where m is the number of quadrats, so the fraction above
is expected to equal 1/m. The ratio of the observed and expected fractions is the Morisita index
[488]
∑ j n j (n j − 1)
M=m
.
(7.1)
n(n − 1)
This index should be close to 1 if the points are independent, greater than 1 if they are clustered,
and less than 1 if they are regular.
Repeating this calculation using different subdivisions of the window into quadrats, and plotting
the Morisita index against the diameter of the quadrats, yields a Morisita index plot. Figure 7.2
shows that this has the ability to distinguish between the three archetypal point patterns in Figure 7.1.
In spatstat the Morisita index plot of a point pattern X is generated by miplot(X).
The Morisita index is closely related to the index of dispersion for quadrat counts. Suppose we
calculate the sample variance of the quadrat counts,
s2 =
1
m−1
m
∑ (n j − n)2
j=1
where n is the average quadrat count, n = n/m. If the point process is Poisson, then the counts
n1 , . . . , nm are observations of independent Poisson random variables with the same, unknown mean
µ . The variance of a Poisson random variable is equal to its mean. A standard index of overdispersion or underdispersion for count variables is the sample variance divided by the sample mean: for
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Figure 7.2. Morisita index plots (on logarithmic scale) for the three patterns in Figure 7.1.
a Poisson distribution this ratio should be approximately equal to 1. Dividing the sample variance
s2 by the sample mean n gives the index of dispersion
I=
s2
m
=
n̄
n(m − 1)
m
n
∑ (n j − m )2 .
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An earlier paper by Greig-Smith [296] had proposed plotting the index of dispersion against quadrat
size. A little algebra shows that these are equivalent: the dispersion index I and Morisita index M
are directly related by I = (m/(m − 1))[(n − 1)M − (n/m − 1)]. As discussed on page 167, the
index of dispersion is also closely related to the χ 2 test of CSR based on quadrat counts. Thus, the
Morisita and Greig-Smith plots are effectively plots of the test statistic for a χ 2 quadrat counting
test of CSR plotted against the size of the quadrats. See [462, 575, 225].
For the purposes of this chapter we will focus on the Morisita index M. Note especially that M
assumes the intensity is homogeneous. If this is not the case, large values of M could arise simply
from spatial inhomogeneity, rather than from some form of correlation between the points.
The Morisita index is not sensitive to subtle differences in spatial scale, because it is based on
subdividing the observation window coarsely into quadrats. This suggests that we look for better
ways to summarise the information about pairs of points.
There is also something unsatisfactory about the theoretical derivation of the Morisita index
given above. That derivation referred only to the homogeneous Poisson point process, and calculated that the Morisita index should be about 1 for that process. But if the data were generated by
another kind of point process, it is unclear (at least from the previous discussion) how to interpret
the value of the Morisita index. Indeed the Morisita index might not even be a well-defined property
of the point process: for example, it might depend on the size of the observation window. Although
these questions can be resolved, the answers are not very simple.
A good statistical index should not only be accessible by simple direct calculation from the data,
but it should have a simple, direct interpretation for the point process which generated the data. In
the rest of the chapter we look for such indices.
7.2.2
Fry plot
More information about spacings in the point pattern can be obtained using a Fry plot or Patterson
plot. Originally developed for crystallography by Patterson [534, 535] this technique was independently reinvented in geophysics by Fry [271, 321].
A Fry plot can be drawn by hand, as follows (see Figure 7.3). First print the point pattern on a
sheet of paper. Take a transparency or sheet of tracing paper, and mark a cross in the middle. Place
the transparency over the printout, so that the cross on the transparency lies on one of the data points.
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Copy the positions of all the other data points onto the transparency in the same relative position.
(Of course some data points may be too far away to be copied onto the transparency, depending on
its size.) Now move the cross to another data point, and repeat the copying process. After every
data point has been visited, the pattern on the transparency is the Fry plot.
(a)
(c)
(d)
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(b)
Figure 7.3. Stages in forming the Fry plot. (a): data point pattern, printed on paper. (b): transparency superimposed on point pattern so that centre of transparency (+) lies above the first data
point. (c): other data points are copied (◦) onto the transparency. (d): final result.
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In mathematical terms the Fry plot is a scatterplot of the vector differences x j − xi between all
pairs of distinct points in the pattern.
Figure 7.4. Fry plots for the three patterns in Figure 7.1.
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In spatstat the Fry plot of a point pattern X is generated by the command fryplot(X)
or plot(frypoints(X)). Additional arguments allow the plot to be restricted to a smaller area
around the origin (0, 0), where the interesting detail is found, or restricted to certain groups of data
points.
Figure 7.4 shows the Fry plots for the three archetypal point patterns in Figure 7.1, restricted to
squares of width 0.6 units. The origin is in the middle of each panel of Figure 7.4 and is indicated
by the ‘+’ symbol, visible only in the left panel.
The origin in the Fry plot represents a typical point of the point pattern, and the dots in the Fry
plot represent the positions of other nearby points, relative to the typical point. In the left panel
of Figure 7.4 there is a clear absence of dots in the middle of the panel, indicating that data points
never come closer to each other than a certain minimum distance. The middle panel of Figure 7.4
shows no obvious pattern, while the right panel shows a higher concentration of dots near the origin,
indicating a clustered pattern. Thus, the Fry plot is easily able to distinguish the three basic kinds
of dependence between points.
Fry plots are implicitly based on the assumption that the underlying process is stationary (Section 5.6.3). Under this assumption, the Fry plot contains essentially all information about correlations in the point process.
Fry plots can be very useful for spotting features of the point pattern which might not otherwise
be obvious, such as discretisation of the coordinates. In geophysics, Fry plots have proven to be
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very useful for inferring mechanical strain in rocks, which is reflected in the shape of an elliptical
‘hole’ in the Fry plot. However, in many other applications, the interpretation of the Fry plot is too
subjective. Fry plots often need to be simplified, reduced, or summarized in order to extract usable
information.
7.3 The K-function
7.3.1
The empirical K-function
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A very popular technique for analysing spatial correlation in point patterns is the K-function proposed1 by Ripley [572].
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Suppose that the primary research question concerns the distance or spacing between points in the
point pattern. It would then be natural to measure the distances di j = kxi − x j k between all ordered
pairs of distinct points xi and x j in the point pattern x under consideration. These distances clearly
capture a great deal of information about the spatial pattern. If the pattern is clustered, many of the
pairwise distances will be small; if the pattern is regular, few of the distances will be small. This
suggests that we might look at a statistical summary of the distances di j , such as the histogram.
We have argued that a good statistical summary of a point pattern should have a simple, direct
interpretation in terms of the point process which generated the data. The histogram of observed
pairwise distances di j is difficult to interpret in this way, because it depends on the shape and size of
the observation window: the same point process, viewed through different windows, yields different
histograms of pairwise distances.
Consider instead the empirical cumulative distribution function of the pairwise distances,
b = fraction of values di j less than r
H(r)
n n 1
=
1 di j ≤ r
∑
∑
n(n − 1) i=1 j=1
(7.2)
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j6=i
defined for each distance value r ≥ 0. Here we use the ‘indicator’ notation: 1 {. . .} equals 1 if the
statement ‘. . . ’ is true, and 0 if the statement is false. The sum is taken over all ordered pairs i, j
of indices which are not equal. The sum of these indicators is simply the number of times that the
statement is true, that is, the number of values di j which are less than or equal to r. The denominator
b is the fraction of pairs for which the
n(n − 1) is the total number of pairs of distinct points, so H(r)
distance is less than or equal to r.
b
Notice that H(r)
is analogous to the Morisita index: both quantities report the fraction of pairs
b
of points which lie close together. The distance argument r in H(r)
defines ‘closeness’, and is
analogous to the size of quadrats in the Morisita index.
b
We can also visualise the calculation of H(r)
using the Fry plot. The dots in the Fry plot are
the vector differences xi − x j between all pairs of distinct points in the point pattern dataset. The
lengths of these vectors are the distances di j . To count the number of distances di j that are less than
or equal to r, we simply draw a circle of radius r, centred at the origin of the Fry plot, and count the
b
number of dots in the Fry plot which fall inside this circle. That is, H(r)
is the fraction of dots in
the Fry plot which fall inside the circle of radius r.
1 There
are similar concepts in statistical physics [531] and astronomy (cf. [445]).
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The contribution from each data point xi to the sum in (7.2) is
ti (r) = ∑ 1 di j ≤ r ,
j6=i
the number of other data points x j which lie closer than a distance r. We might call this the number
of r-neighbours for the point xi . Equivalently ti (r) is the number of data points which fall inside a
circle of radius r centred at xi , not counting xi itself. Then
b =
H(r)
n
1
1
ti (r) =
t(r)
∑
n(n − 1) i=1
n−1
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where t(r) = (1/n) ∑i ti (r) is the average number of r-neighbours per data point.
This is an important clue that the quantity we really
want to estimate is the average number of r-neighbours
of a typical random point. Figure 7.5 shows an intermediate stage in the calculation of t(r). The number of
2
3
r-neighbours is shown for each of several data points.
The average of these numbers for all data points is t(r).
6
The average number of r-neighbours of a data point
will depend on the overall average density of points in
3
the dataset. In a completely random pattern, we would
expect t(r) to be about λ π r2 , since t(r) counts the num4
0
ber of points falling in a circle of radius r which has
area π r2 . In order to be able to compare datasets with
different numbers of points, it makes sense to standardise t¯(r) by dividing by λ . Since the maximum numFigure 7.5. Counting r-neighbours. In ber of neighbours of any data point is n − 1, it may be
each circle, the numeral shows the value more appropriate to divide by λ̃ = (n − 1)/|W |, where
of ti (r) for the data point xi at the centre n is the number of points and |W | is the area of the obof the circle.
servation window. The result of this standardisation is
b
t(r)/((n − 1)/|W |) = |W | H(r).
b
The function |W | H(r)
is the standardised average number of r-neighbours of a typical data
point. In order to be fully able to compare datasets observed in different windows, we also need to
take account of edge effects as explained in Section 7.4. This leads to a slight modification of the
b
function |W |H(r),
called the empirical K-function,
b =
K(r)
|W | n n 1 di j ≤ r ei j (r)
∑
∑
n(n − 1) i=1 j=1
(7.3)
j6=i
where ei j (r) is an edge correction weight described in Section 7.4.
b
In summary, the empirical K-function K(r)
is the cumulative average number of data points
lying within a distance r of a typical data point, corrected for edge effects, and standardised by
dividing by the intensity. The standardisation and edge correction make it possible to compare point
patterns with different numbers of points, observed in different windows.
Figure 7.6 displays the empirical K-functions for the three archetypal point patterns of Figure 7.1, showing that the empirical K-function clearly has the ability to discriminate between the
three basic kinds of interpoint dependence. The empirical K-function for the clustered pattern lies
above the empirical K-function for a completely random pattern, which in turn lies above the empirical K-function for a regular pattern. This is equivalent to saying that, after adjusting for intensity,
a typical point in the clustered pattern has more close neighbours than a typical point in the completely random pattern, which in turn has more close neighbours than a typical point in the regular
pattern.
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independent
regular
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r
Figure 7.6. Empirical K-functions for the three patterns in Figure 7.1. Solid line: regular pattern.
Dotted line: independent pattern. Dashed line: clustered pattern.
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b is a stanWhen interpreting graphs of the empirical K-function, it helps to remember that K(r)
dardised or relative quantity, rather than a direct physical quantity. The empirical K-function of a
completely mapped forest gives us, for each r, the average number of neighbour trees lying within
distance r of a typical tree, divided by the density of the forest in trees per unit area. The values
b
of K(r)
are expressed in units of (number)/(number/area) = area. Standardisation makes it
possible to compare the degree of regularity or clustering in forests with different average densities
of trees. To recover the physically meaningful average number of neighbours t(r), we would need
b by λ to obtain t(r).
to know the forest density λ , and multiply K(r)
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Warning: using the K-function implicitly assumes that the point process has homogeneous
intensity. See Section 7.3.5.
7.3.2
The true K-function of a point process
b
The empirical function K(r)
is a summary of the pairwise distances in the point pattern dataset,
normalised to enable us to compare different datasets. But the key question about any summary
statistic for a point pattern is what it means for the point process which generated the pattern.
The K-function of a point process X will be defined as the expected number of r-neighbours of
a typical point of X, divided by the intensity λ . For this we need to assume2 that X is stationary
(see Section 5.6.3: the distribution of X is the same as the distribution of the shifted process X + v,
for any vector v). This implies that X has homogeneous intensity λ . We may then define
K(r) =
1 E number of r-neighbours of u X has a point at location u
λ
(7.4)
for any r ≥ 0 and any location u. Since the process is stationary, this definition does not depend
on the location u. On the right-hand side of (7.4), the symbol ‘|’ indicates that this is a conditional
expectation. Intuitively, we assume there is a random point of X at the location u; given this, we
2 Slightly
weaker assumptions are enough, and these will be stated below.
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find the expected number of other points of X lying within a distance r; and finally we divide by the
intensity λ , to obtain K(r).
Extending the notation ti (r), let us define for any spatial location u
n(x)
t(u, r, x) =
∑1
j=1
0 < ku − x j k ≤ r ,
(7.5)
the number of points in the point pattern x that lie within a distance r of the location u, but not at u
itself.
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Definition 7.1. If X is a stationary point process, with intensity λ > 0, then for any r ≥ 0
K(r) =
1 E t(u, r, X) u ∈ X
λ
(7.6)
does not depend on the location u, and is called the K-function of X.
Explicit formulae for the K-function have been derived for a few point process models. For the
homogeneous Poisson point process (CSR), since the points are independent, intuitively speaking,
the presence of a random point at the location u will have no bearing on the presence of points at
other locations, so
E t(u, r, X) u ∈ X = E [t(u, r, X)] .
But t(u, r, X) is the number of points of X falling in the disc b(u, r) of radius r centred at u. The expected number of such points is λ × |b(u, r)| = λ π r2 . Dividing by λ shows that, for a homogeneous
Poisson process,
(7.7)
Kpois (r) = π r2
7.3.3
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regardless of the intensity. This calculation is for two dimensions; for the case of three dimensions
see Chapter 15.
Use of the empirical K-function
Visual inspection of empirical K-function
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To study correlation in a point pattern dataset, assuming the intensity is homogeneous, we can plot
b calculated from the data, together with the theoretical K-function of
the empirical K-function K(r)
the homogeneous Poisson process Kpois (r) = π r2 , which serves as the benchmark of ‘no correlation’.
Figure 7.7 shows this graphic for each of the three archetypal patterns in Figure 7.1.
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0.00 0.05 0.10 0.15 0.20 0.25
0.00 0.05 0.10 0.15 0.20 0.25
r
r
r
Figure 7.7. Empirical K-function (solid lines) for each of the three patterns in Figure 7.1, and the
theoretical K-function for a Poisson process (dashed lines).
Figure 7.7 shows that this graphic easily detects the presence and type of correlation between
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points in a point pattern. In the left panel, the curve for the empirical K-function (solid lines) is lower
b < Kpois (r), indicating
than the theoretical curve for a completely random pattern (dashed lines), K(r)
that a typical point in this pattern has fewer neighbours than would be expected if the pattern were
completely random. This is consistent with a regular point process. Similarly in the right panel,
b > Kpois (r), indicating that a typical
the empirical curve is higher than the theoretical curve, K(r)
point has more neighbours than would be expected if the pattern were completely random; this is
consistent with clustering. The K-function is a powerful tool for investigating point patterns, but
b > Kpois (r), the
we need to remember that ‘correlation is not causation’. If analysis shows that K(r)
careful scientist will not say that this ‘indicates’ clustering, but that it is ‘consistent with’ clustering,
or that it indicates ‘positive association’ between points. See Section 7.3.5 for more discussion.
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Transformation of K
The centred version of the K-function is K(r) − Kpois (r) = K(r) − π r2 . This can be useful for
classifying a point pattern as random, clustered, or regular, because the function is zero if the point
pattern is completely random. It is less useful for other purposes.
A commonly used transformation of K proposed by Besag [103] is the L-function
r
K(r)
(7.8)
L(r) =
π
L (r )
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L (r )
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L (r )
0.00 0.05 0.10 0.15 0.20 0.25
which transforms the theoretical Poisson K-function Kpois (r) = π r2 to the straight line Lpois (r) = r,
making visual assessment of the graph much easier. Figure 7.8 shows the empirical L-functions for
the three archetypal point patterns. The square root transformation also approximately stabilises the
variance of the estimator (that is, the variance of the empirical function b
L(r) is roughly constant as
a function of r), making it easier to assess deviations.
0.00 0.05 0.10 0.15 0.20 0.25
0.00 0.05 0.10 0.15 0.20 0.25
0.00 0.05 0.10 0.15 0.20 0.25
r
r
r
Figure 7.8. Empirical L-function (solid lines) for each of the three patterns in Figure 7.1, and the
theoretical L-function for a Poisson process (dashed lines).
Besag’s L-function has become such a popular transformation of Ripley’s K-function that many
writers in applied fields, and some software packages, apply the transformation without mentioning
it. They plot L(r) against r, or the centred version L(r) − r against r, but still call it Ripley’s Kfunction. This is not advisable because K and L are not equivalent in some contexts. When we say
K(r) we shall always mean K(r).
Statistical inference
Options for formal statistical inference about the K-function include confidence intervals and hypothesis tests (the latter associated with simulation envelopes). These ideas are illustrated in Figure 7.9 using the clustered pattern in the right panel of Figure 7.1.
The left panel of Figure 7.9. shows a confidence interval for the true K-function of the point pat-
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tern. The right panel shows an acceptance interval (or ‘non-rejection’ interval) for testing whether
the pattern is completely random.
It is very important to understand the difference between these two techniques. A confidence
interval is designed to contain the true value of the target quantity with a specified degree of confidence. It is centred around an estimated value of the target quantity, and its width is an indication of
the precision of the estimation. In the left panel of Figure 7.9, the shaded region is a pointwise 95%
confidence interval for the true K-function of the clustered pattern; the shading is centred around
the estimated K-function; the width of the shaded region reflects the precision of the estimate. Section 7.7 explains how to construct confidence intervals for the K-function.
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On the other hand, an acceptance interval (or ‘non-rejection interval’) contains the hypothesised value of the target quantity. It is the range of values that are deemed to be not significantly
different from the hypothesised value. The width of the acceptance interval reflects the inherent variability of the observations when the hypothesis is true. In the right panel of Figure 7.9, the shaded
region is the acceptance interval for a formal test, with significance level 0.05, of the hypothesis
that the pattern is completely random. The acceptance interval is centred around the K-function of a
completely random point pattern, Kpois (r) = π r2 , and the width of the interval reflects the variabilb
b
ity of K(r).
For a fixed value of r, the test rejects the null hypothesis of CSR if the value of K(r)
lies outside this acceptance interval. Section 7.8 explains how to construct acceptance intervals for
testing CSR. More explanation and important caveats about hypothesis tests are given in Chapter 10.
Figure 7.9. Tools for formal statistical inference, demonstrated on the clustered pattern in the
right panel of Figure 7.1. Left: 95% confidence intervals (shaded) for the true value of the Kfunction, obtained using Loh’s bootstrap (function lohboot, Section 7.7.3). Right: acceptance
region (shaded) for a hypothesis test of complete spatial randomness, with significance level 5%,
using the envelopes of the K-functions of 39 simulated realisations of CSR. Generated using the
function envelope (Sections 7.8 and 10.7).
The K-function as a guide for building models
The K-function can serve as a guide for building a point process model of the phenomenon being studied. Chapter 12 presents one class of models which can be fitted directly to point pattern
data using the K-function. More commonly the K-function is only a rough guide to the required
behaviour of the model.
A point process model of a natural phenomenon could include several stages in which points are
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randomly added, combined, displaced, or removed. In this regard there are two important properties
of the K-function.
Firstly the K-function is invariant under random thinning. Suppose X is a stationary point
process, and Y is the point process obtained by independent random thinning — randomly deleting
or retaining each point of X, with a constant probability p of retaining each point (Section 5.3.3).
Then the K-function of Y is identical to the K-function of X.
Secondly there is the effect of superposition. Suppose X and Y are two independent, stationary
point processes, with intensities λX , λY and K-functions KX (r), KY (r). Superimposing these two
processes gives a stationary point process with intensity λ = λX + λY and K-function
(7.9)
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K(r) = p2X KX (r) + p2Y KY (r) + 2pX pY π r2
where pX = λX /λ and pY = λY /λ are the relative proportions of points coming from X and Y. The
centred K-functions are more simply related:
(K(r) − π r2 ) = p2X (KX (r) − π r2 ) + p2Y (KY (r) − π r2 ).
7.3.4
(7.10)
Estimating the K- and L-functions in spatstat
The spatstat function Kest computes several estimates of K(r) using different edge corrections.
It also returns the ‘benchmark’ value π r2 which is the theoretical value for K(r) for a homogeneous
Poisson process.
> K <- Kest(ells)
> Ki <- Kest(ells, orretion="isotropi")
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Here ells is Ripley’s [572] biological cells dataset (shown on page 93) which we shall use frequently as an example.
The argument orretion specifies which estimate or estimates will be computed by Kest.
Options include "isotropi" for Ripley’s isotropic correction, "translation" for the translation correction, "rigid" for the rigid motion correction, "border" for the border correction, and
"none" for the uncorrected estimate. These are explained in Section 7.4. Any number of edge
corrections may be selected. Specifying a single edge correction will reduce computation time,
especially in simulation experiments.
Additionally Kest, like all standard summary functions in spatstat, recognises the options
"best" (representing the estimate with the best statistical performance regardless of computational
cost) and "good" (the estimate with the best statistical performance for reasonable computational
cost).
If no orretion argument is given, the default is to compute the isotropic, translation, and
border corrections, unless the point pattern contains more than nlarge points, when only the border
correction will be computed. The default threshold is nlarge = 3000 points. Setting nlarge=Inf
will suppress this behaviour.
The spatstat function Lest computes estimates of L(r) directly from point pattern data.
> L <- Lest(ells)
The arguments of Lest are identical to those of Kest.
See Section 7.5 for a detailed explanation of how to plot and manipulate estimates of the Kfunction and other summary statistics. A previously computed K-function can be converted to the
L-function using the syntax L <- with(K, sqrt(./pi)), explained on page 224, or plotted as
an L-function using the syntax plot(K, sqrt( /pi)~r), explained on page 221.
.
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Spatial Point Patterns: Methodology and Applications with R
Caveats about the K-function
The use of the K-function for analysing point patterns has become established across wide areas
of applied science, following Ripley’s influential paper [572] and many subsequent textbooks [190,
221, 225, 661, 575, 576, 638]. Useful and powerful as this methodology is, there is an unfortunate
tendency to apply it uncritically and to neglect other methods of analysis.
7.3.5.1 K-function assumes homogeneity
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The K-function is defined and estimated under the assumption that the point process is stationary.
b
If the process is not stationary, deviations between the empirical and theoretical functions (e.g., K
and Kpois ) are not necessarily evidence of interpoint interaction, since they may also be attributable
to variations in intensity [83].
Figure 7.10 shows a realisation of an inhomogeneous Poisson process, and its empirical Kfunction. The plot of the K-function shows that, on average, a point in this pattern has more rneighbours than would be expected for CSR. However, this happens because there is a higher density
of points in one corner of the window. The points are positively associated, but not because they are
clustered in the usual sense.
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0.1
0.2
K pois (r )
0.0
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K (r )
0.3
K^ iso (r )
0.00
0.05
0.10
0.15
0.20
0.25
r
Figure 7.10. The K-function is fooled by spatial inhomogeneity. Left: Inhomogeneous Poisson
point pattern. Right: Empirical K-function.
7.3.5.2 Correlation is not causation
There are many possible causes and mechanisms of correlation between points. True spatial regularity could be the result of ecological processes (territorial behaviour, competition for resources),
physical forces (repulsion between objects), or human intervention. Apparent spatial regularity can
occur as an artefact of the treatment of data, for example, if biological cells of appreciable size are
treated as ideal points, or if spatial coordinates are discretised. Spatial clustering does not imply that
the points are organised into identifiable ‘clusters’; merely that they are closer together than would
be expected for a completely random pattern. True spatial clustering could be the result of biological processes (reproduction, contagion), physical forces (electrostatic or magnetic attraction),
or space-time history (clustering of meteorite impacts). Apparent clustering can occur if parts of
a spatial pattern are obliterated (e.g., native forest partially destroyed by fire) or if objects of one
type are physically repelled by another type of object. Spatial inhomogeneity is often mistaken for
spatial clustering as we mentioned above.
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7.3.5.3 Lack of correlation does not prove independence
> Xell <- rell(nx=15)
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Correlation is a summary measure of stochastic dependence, but absence of correlation does not
necessarily indicate independence. Examples are well known in elementary statistics.
Similarly, there exist point processes whose K-functions are equal to π r2 and yet the processes
are not Poisson processes (so that there is dependence amongst the points). Therefore the K-function
does not completely characterise the point process.
An example is the cell process [69]. Space is divided into equal tiles or cells; in each cell we
place a random number N of points, according to a probability distribution whose variance is equal
to its mean (EN = var N); the points are positioned independently and uniformly within the cell.
The cell process has exactly the same theoretical K-function as the homogeneous Poisson process,
but is manifestly different from a Poisson process. The left panel of Figure 7.11 shows a realisation
of the cell process, generated by the spatstat function rell:
0.20
The right panel shows the empirical K-function, which falsely suggests that the process is Poisson.
In fact the cell process would defeat any technique based on second moments, because all secondmoment quantities for the cell process are identical to those for the Poisson process. Ripley [576,
p. 23] commented: “It is a measure of the success of second-order methods that they have been
mistakenly assumed to be all-powerful!”
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0.05
0.10
K pois (r )
0.00
K (r )
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0.15
K^ iso (r )
0.00
0.05
0.10
0.15
0.20
0.25
r
Figure 7.11. Realisation of Baddeley-Silverman cell process (Left) and its empirical K-function
(Right).
7.3.5.4 Spatial scale of interaction
Summary functions like Ripley’s K-function convey information across a range of spatial scales.
This is an important motivation for using empirical functions, rather than simple numerical summary
statistics, and for displaying them graphically.
Many researchers use a graph of the K-function to infer ‘the’ scale of spatial interaction in a
point pattern. This scale is often estimated by reading off the position where the empirical function
lies furthest away from the theoretical Poisson value, or furthest outside the region enclosed between
the simulation envelopes.
This interpretation is not correct for several reasons. Firstly, ‘the scale of interaction’ is not a
well-defined concept for point processes in general. It is only meaningful for certain point process
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models, such as Markov point processes (Chapter 13) and some Neyman-Scott cluster processes
(Chapter 12).
Secondly, even for point processes which have a well-defined scale of interaction, it is not always
true that the greatest deviation in the K-function occurs when r is equal to the scale of interaction.
Examples are given in [40, p. 487]. The K-function reflects correlation between pairs of points, not
direct dependence. Dependence between points at one scale can give rise to correlation between
points at another scale.
Thirdly, the K-function is cumulative: K(r) accumulates contributions from all distances less
b > π r2 for the distance r = 10
than or equal to r. If a pattern of emergent seedlings yields K(r)
metres, this does not necessarily indicate that seedlings are clustered at distances of 10 metres. A
plausible explanation is that seedlings are organised in clusters at a much smaller spatial scale, and
the cumulative effect is still evident at 10 metres. An alternative, non-cumulative summary statistic
is the pair correlation function (Section 7.6). The K-function is optimal for detecting interpoint
interaction that occurs equally at all distances up to a certain maximum distance r.
7.4 Edge corrections for the K-function∗
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This section gives more detail about the edge correction techniques used in estimating the Kfunction of a point process.
Most users of spatstat will not need to know the theory behind edge correction, the details of
the techniques, or their relative merits. So long as some kind of edge correction is performed (which
happens automatically in spatstat), the particular choice of edge correction technique is usually
not critical.
However, there is a danger that the implementations of edge corrections in some software packages may be incorrect, which could cause substantial bias. There are certainly some misunderstandings about edge corrections that have crept into the applied literature. This section attempts to set
the record straight.
For advanced users, understanding the assumptions behind edge correction is helpful in appreciating potential weaknesses of the analysis, and in resolving discrepancies between results. Edge
corrections are reviewed in [576, chap. 3], [59, 355].
7.4.1
Estimation without edge effects
We recall that di j = kxi − x j k is the distance between a pair of distinct data points xi , x j , and
n(x)
t(u, r, x) =
∑1
j=1
0 < ku − x j k ≤ r
is the number of r-neighbours of the location u. An alternative expression for the K-function, that
does not involve conditional expectation, is
K(r) =
E ∑x∈X∩B t(x, r, X)
λ En(X ∩ B)
(7.11)
holding for a stationary point process X, and for any bounded region B with area |B| > 0. Thus
λ K(r) is the expected number of r-close pairs of points in which the first point falls in B, divided
by the expected number of points falling in B.
∗ Starred sections contain advanced material, and can be skipped by most readers.
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Correlation
A straightforward way to estimate the K-function is suggested by the representation in equation
(7.11). Given a point pattern dataset x, we simply replace the numerator and denominator of (7.11)
by data-based estimates:
t(x , r, x)
e = ∑xi ∈x∩B i
K(r)
.
(7.12)
λ n(x ∩ B)
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The numerator is the total number of r-neighbours of all random points falling in B (an arbitrary
set with nonzero area), and in the denominator, n(x ∩ B) is the number of points falling in B, while
λ = n(x ∩ B)/|B| is an estimate of λ .
Figure 7.12. Counting, without edge effects, the number of neighbours of each wildflower within a
sampling frame (black rectangle) in a field of wildflowers.
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Figure 7.12 sketches how this estimator could be applied in practice. In a homogeneous field of
wildflowers, we have laid out a sampling frame B (black square). Visiting one of the flowers inside
the sampling frame, we push a thin stake into the soil near the flower (crosshairs), tied to a string
exactly one metre long. Pulling the string taut, we describe a circle around the stake (dashed lines),
and count how many other flowers are inside the circle (black dots) regardless of whether they lie
inside or outside the frame. For the example in Figure 7.12 the count is 14. We have just counted
the number of r-neighbours t(xi , r, x) for one flower xi , for the distance r = 1 metre.
Repeating this process for each flower inside the study area would give us the neighbour counts
t(xi , r, x) for flowers x1 , . . . , xn , say, where n is the number of flowers in the study area. The average
observed neighbour count is t(r) = (1/n) ∑i t(xi , r, x). Dividing by the estimated intensity λ = n/|B|,
where |B| is the area of the study frame B, gives us an estimate of K(r) as in (7.12).
7.4.2
Edge effects
The situation sketched in Figure 7.12 is unusual, because we were able to look outside the sampling
frame. In most research studies, this information is lost; points are recorded only if they fall inside
the study region; the situation is more like Figure 7.13.
If points are observed only inside a window W , then the estimator (7.12) with B = W is not
feasible. The number of points inside a circle of radius r, centred on a point of the process inside
W , is not observable if the circle extends outside W . This is an edge effect problem.
It might be tempting to ignore this problem and use the ‘uncorrected’ empirical function
bun (r) =
K
|W | n n b
∑ ∑1 di j ≤ r = |W |H(r),
n(n − 1) i=1
j=1
(7.13)
j6=i
the counterpart of (7.3) without the weighting terms ei j (r). However, a simple experiment shows
that (7.13) is severely biased as an estimator of K(r). We generate a completely random point
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Figure 7.13. Edge effect problem for estimation of the K-function. If we can only observe the points
inside a window W (solid lines), then the number of points inside a circle (dashed lines) of radius r,
centred on a point of the process inside W , is not known if the circle extends outside W . The number
of points observed inside the circle (i.e. counting only points within the window) is typically less
than the true number.
0.00
0.05
M
0.10
K (r )
0.15
0.20
pattern, according to a uniform Poisson process with intensity λ = 100 in the unit square. The
bun (r) is plotted in Figure 7.14 together with the correct K-function K(r) =
uncorrected function K
2
π r . The uncorrected function is clearly an underestimate of the correct K-function.
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0.00 0.05 0.10 0.15 0.20 0.25
r
bun (r)
Figure 7.14. Bias in estimating K(r) due to edge effects. Solid lines: uncorrected estimate K
2
for a completely random point pattern. Dashed lines: correct K-function K(r) = π r .
The edge effect bias can be predicted for a given window W using the function distdf, which
computes the true cumulative distribution function H(r) of the distance between two independent
random points in the window.
7.4.3
Border correction
An edge correction is a strategy for eliminating the edge effect bias. Edge corrections are surveyed
in [576, chap. 3], [59], [355, sec. 4.3.3].
One simple strategy is the border method. When estimating K(r) for a particular distance r, we
restrict attention to cases where the circle of radius r lies entirely inside the window, so that the edge
effect does not occur. See Figure 7.15.
In the numerator and denominator of (7.12) we restrict the summation to data points xi for which
b(xi , r) lies entirely inside W . For such points, t(xi , r, X ∩ W ) = t(xi , r, X), so the true number of r-
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Figure 7.15. Border method of edge correction for the K-function. When estimating K(r), distances
di j are measured only from points xi lying at least r units away from the boundary. These are the
points falling inside W⊖r (shaded region).
neighbours is observable. If d(u, ∂ W ) denotes the shortest distance from a location u to the window
boundary ∂ W , then we are restricting attention to data points xi which satisfy d(xi , ∂ W ) ≥ r. These
are the data points which fall in the eroded set
W⊖r = W ⊖ b(0, r) = {u ∈ W : d(u, ∂ W ) ≥ r}
(7.14)
M
consisting of locations in W that are at least r units away from the boundary ∂ W . The eroded set is
shaded in Figure 7.15.
Thus we estimate K(r) by the border correction estimate
t(x , r, x) ∑ni=1 1 {bi ≥ r} t(xi , r, x)
bbord (r) = ∑xi ∈x∩W⊖r i
K
=
λ n(x ∩W⊖r )
λ ∑ni=1 1 {bi ≥ r}
(7.15)
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where bi = d(xi , ∂ W ) is the distance from the data point xi to the window boundary, and λ is an
estimate of the intensity, usually λ = n(x ∩W )/|W |.
The estimate (7.15) is well defined so long as the denominator is non-zero, that is, so long
as r < maxi bi . The maximum possible value of r for which (7.15) can ever be computed is3 the
inradius of W , the radius of the largest disc contained in W , since this is the maximum possible
value of d(·, ∂ W ).
The border correction estimate (7.15) can be justified as a data-based estimate of the right-hand
side of (7.11) taking B = W⊖r . This suggests that it will have reasonable statistical properties in
sufficiently large datasets. The numerator of (7.15) is an unbiased estimator of the numerator of
(7.11), while the denominator of (7.15) is the product of two terms which are unbiased estimators
of the corresponding terms in the denominator of (7.11). In large datasets, the border-correction
estimate is consistent and approximately unbiased, under reasonable assumptions.
Like many estimators in spatial statistics, the empirical K-function in (7.15) is a slightly biased
estimator of the true K-function. It is composed of several terms, each of which is an unbiased
estimator of a corresponding theoretical quantity. This implies that the overall bias of the empirical
K-function will be small in large samples. Another way to write equation (7.11) is
λ 2 K(r) =
1
E ∑ t(x, r, X)
|B| x∈X∩B
(7.16)
3 Here and throughout the discussion we assume W is a topologically regular set, meaning that it is the closure of its
interior.
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0.10
7.4.4
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0.05
K (r )
0.15
0.20
so that λ 2 K(r) is the expected number of r-close pairs of points with the first point falling in B,
divided by the area of B. In essence it is really λ 2 K(r) that we need to estimate.
The border correction estimate of the Kfunction is relatively simple to implement in
software for any shape of window, and is fast
to calculate. However, in small datasets, it can
be inaccurate when compared to other methods,
because it discards substantial amounts of data.
For example, if W is the unit square and r = 0.1,
the eroded window W⊖r is a square of side 0.8
and area 0.64 so that almost 40% of the data is
being discarded in order to estimate K(0.1). In
small datasets or in unusual situations, the graph
bbord (r) can have an
of the estimated function K
erratic trajectory, as shown in Figure 7.16. For
large values of r, the denominator of (7.15) is
0.00 0.05 0.10 0.15 0.20 0.25
small, magnifying the effect of discrete jumps in
r
the numerator and denominator. Although the
Figure 7.16. Border correction estimate of the true K-function must be an increasing function
K-function (solid lines) for 20 uniformly random of r, the border correction estimate need not be
so.
points in the unit square.
When the number of data points is large, the
border method is usually preferable to more computationally intensive methods, because the interesting distances r are small, and the loss of statistical efficiency in the border method is negligible.
The border method is a useful general-purpose remedy for edge effects in many spatial problems.
It is related to the ‘local knowledge principle’ of mathematical morphology [622, 65].
Isotropic correction
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An alternative approach to edge effects is to regard them as a form of sampling bias.
Visualise the entire point process X extending throughout two-dimensional space. Consider a
pair of points x, x′ from X, and assume that x falls in the observation window. Given the location
of the first point x and the distance d = kx − x′ k, the second point x′ must lie somewhere on the
circle b(x, d) of radius d centred at x. In general, only part of this circle lies inside the window: see
Figure 7.17.
If the point process is also isotropic (statistically invariant under rotation, Section 5.6.3), then
roughly speaking, the second point x′ is equally likely to lie anywhere on this circle. The probability
that x′ falls inside W is the fraction of length of the circle lying within W ,
p(x, d) =
ℓ(W ∩ ∂ b(x, d))
2π d
where ℓ denotes length. As d increases, the probability p(x, d) typically decreases. This gives rise
to a sampling bias: larger distances are less likely to be observed.
A standard strategy for correcting sampling bias is the Horvitz-Thompson estimator [341, 58,
173] introduced on page 181. This strategy can be applied to estimators of the K-function. If each
pair of points xi , x j is weighted by the reciprocal of the probability p(xi , di j ), we obtain Ripley’s
isotropic correction estimator [572, 569]
biso (r) = 1
K
λn
n
n
∑ ∑1
i=1 j=1
j6=i
di j ≤ r
1
.
p(xi , di j )
(7.17)
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Figure 7.17. Calculation of sampling bias for the isotropic correction. The contribution from a pair
of data points xi , x j is determined by drawing a circle through x j centred at xi . The fraction p of
the circle’s perimeter which falls inside the observation window (grey line) is measured. The edge
correction weight for this pair of points is 1/p.
Some authors state, incorrectly, that Ripley’s isotropic correction uses the fraction
of area rather than the fraction of length of the circle. This error may have found its way into
computer code.
Warning:
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The isotropic-correction estimate is a non-decreasing function of r, unlike the border-correction
estimate.
The Horvitz-Thompson principle is valid provided each item in the population has a non-zero
probability of being sampled. In the isotropic correction, it is possible to have p(x, d) = 0 when
d is large enough. Let Rbiso be the smallest distance for which there is some location u in W (not
2
biso (r) is an unbiased estimator of λ 2 K(r) for
necessarily a data point) where p(u, d) = 0. Then λ K
all r < Rbiso . The distance Rbiso is the circumradius of W , the radius of the smallest circle containing
W (or the minimum of the circumradii of the connected components of W , if W consists of several
pieces). If W is a rectangle, Rbiso is equal to half the diagonal of W .
The isotropic estimator can be modified to work for longer distances. For a given distance r, let
W ∗ (r) be the subset of W consisting of locations u where p(u, r) > 0. Then the modified estimator
of Ohser [518] is
n n 2π di j
|W |
biso∗ (r) = 1 ∑ ∑1 di j ≤ r
K
ℓ(W ∩ ∂ b(xi , di j )) |W ∗ (di j )|
λ n i=1 j=1
(7.18)
j6=i
2
biso∗ (r) is an unbiased estimator of λ 2 K(r) for all r < Rbiso∗ . Here Rbiso∗ is the smallest
and λ K
distance R such that |W ∗ (R)| = 0, which is equal to the diameter of W if W is a connected set. The
extra factor in (7.18) is the reciprocal of the fraction |W ∗ (di j )|/|W | of window area occupied by
W ∗ (di j ). Of course (7.18) agrees with (7.17) for r < Rbiso so this calculation is only required when
Rbiso ≤ r < Rbiso∗ .
biso (r), is an unbiased estimator of
For any r < Rbiso , the double sum in (7.17), namely λ nK
2
λ |W |K(r).
The Ohser modified isotropic correction provides estimates of K(r) for much larger distances
r than are possible with the border correction. However, at large distances, the variance of the
estimator increases rapidly. This is a well-known problem with the Horvitz-Thompson estimator: if
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7.4.5
Translation correction
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the probability of sampling an item is very small, it will be given a very large weight if it is sampled,
so its contribution has large variance.
A pragmatic solution is to restrict the range of r values, and to truncate the edge correction
weight in (7.18) so that it never exceeds a specified upper limit. This is the procedure used in
spatstat.
Stein [625, 626] proposed a more satisfactory solution, in which contributions to the estimator are downweighted if they have large variance. Future versions of spatstat will include this
estimator.
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Another approach to edge correction can be followed if we are not willing to assume the point
process is isotropic, or if the computations required for the isotropic correction are too intensive.
Visualise a stationary point process X (a ‘homogeneous field of wildflowers’). Stationarity
means that if we apply any translation (shift) vector s to the entire point process, the shifted process
X + s is statistically equivalent to X. Focus on a particular pair of random points x and x′ in X. If X
is shifted to X + s, the points x, x′ are shifted to new positions x + s and x′ + s. They are in the same
relative position: that is, the vector difference v = x′ − x from x to x′ remains unchanged.
We may imagine that x and x′ are joined by an arrow. When X is shifted to X + s, the arrow’s
length and direction (given by v = x′ − x) remain fixed, but the arrow’s position is shifted by the
vector s.
An arrow will be observed inside the window W only when both its endpoints x, x′ fall in W .
Intuitively it is clear that a shorter arrow is more likely to fall inside W than a longer arrow, and this
gives rise to a sampling bias: the observed arrows are a biased sample of all arrows, with the bias
favoring shorter arrows.
Consider the possible positions of the starting point x for which both endpoints x and x′ = x + v
fall inside W . Notice that x + v ∈ W if and only if x belongs to W − v, a copy of the window W
shifted by the vector −v. Both endpoints of the arrow fall inside W if and only if x ∈ W ∩ (W − v).
The region W ∩ (W − v) is shaded in Figure 7.18. It is the intersection of the window W with a
shifted copy of itself. The probability that the starting point x falls in the shaded region W ∩ (W − v)
is related to the area of this region.
Figure 7.18. Sampling bias calculation for the translation edge correction. An arrow joining a pair
of random points will be observed only if both endpoints fall in the window W . The shaded region
shows the possible locations for the start of the arrow which guarantee that both ends of the arrow
will be observed.
Applying the Horvitz-Thompson principle leads to the translation correction [518, 520] estima-
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Correlation
tor of the K-function,
n n |W |
btrans (r) = 1 ∑ ∑1 di j ≤ r
.
K
|W ∩ (W − (x j − xi ))|
λ n i=1 j=1
(7.19)
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In this estimator, each pair of data points (xi , x j ) is weighted by the reciprocal of the fraction of
window area in which the first data point xi could be placed so that both points xi , x j would be
observable (assuming their relative positions were held fixed). This effectively compensates for the
sampling bias in observing such pairs.
The edge correction weight in (7.19) can be calculated using simple geometry if W is a rectangle. Otherwise, the most efficient algorithm computes the set covariance function CW (v) =
|W ∩ (W − v)| for all vectors v on a fine grid using the Fast Fourier Transform (because C is the
convolution of two indicator functions) and then extracts the values CW (x j − xi ) for each pair of
points.
The Horvitz-Thompson principle gives an unbiased estimator provided that each item in the
2
btrans (r) is an unbiased
population has a non-zero probability of being sampled. Consequently λ K
estimator of λ 2 K(r) for all r ≤ R, where R is the length of the shortest vector v such that CW (v) = 0.
If W is a rectangle, then R is the length of the shortest side.
The translation correction provides estimates of K(r) for larger distances r than are possible
with the isotropic correction, but again suffers from inflated variance at these large distances. Again
the pragmatic solution used in spatstat is to restrict the range of r values, and to truncate the edge
correction weight in (7.19) so that it never exceeds a specified upper limit.
If the point process is assumed to be isotropic as well as stationary (Section 5.6.3), the translation
correction can be modified to give the rigid motion correction [468, 402, 520]
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n n |W |
brigid (r) = 1 ∑ ∑1 di j ≤ r
K
CW (di j )
λ n i=1 j=1
(7.20)
j6=i
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where CW is the rotational average of CW , that is, CW (r) is the average value of CW (v) over all
vectors v of length kvk = r.
7.4.6
Discussion
Edge corrections are discussed and compared in [576, chap. 3], [355, sec. 4.3.3], [59, 640]. The
choice of edge correction depends partly on the size of dataset. In small datasets (say, fewer than
100 points) statistical performance is very important, and the methods of choice are the HorvitzThompson style weighted edge corrections (translation, isotropic or rigid motion correction); the
border method is too imprecise. In moderately large datasets (say 1000 to 10,000 points) the border
method performs satisfactorily. In huge datasets no edge correction is necessary, and it is computationally efficient to avoid edge correction. The spatstat algorithm for orretion="none" will
handle datasets containing many millions of points.
The choice of edge correction does not, in most instances, seem to be very important, as long as
some edge correction is applied. (Although it must be applied correctly!) Discrepancies between the
results of different edge corrections often tend to indicate unusual features in the data, and usually
suggest that the assumption of stationarity is violated.
The accuracy of estimators of the K-function is also affected by the choice of estimator for
the squared intensity λ 2 . Substituting the natural estimator of λ does not necessarily lead to the
b is the ratio of an estimator of λ 2 K(r) divided by an
most efficient estimator of K(r). Because K(r)
2
estimator of λ , positive correlation between the numerator and denominator will generally reduce
220
Spatial Point Patterns: Methodology and Applications with R
b
the variability of K(r).
Various options for the denominator are canvassed in [355, sec. 4.3.3]. One
example favoured in astronomy [317, 401] is
b
λS (r) =
1
∑ p(xi , r)
CW (r) i
(7.21)
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where, as above, p(u, r) is the length fraction of the circle of radius r centred at location u that lies
inside the window W , and Cw (r) is the rotational average of the set covariance function.
7.5 Function objects in spatstat
7.5.1
Objects of class "fv"
An object of class "fv" (‘function value table’) is a convenient way of storing and plotting several
different estimates of the same function.
The value returned by Kest is an object of class "fv":
> KC <- Kest(ells)
> KC
SA
M
Funtion value objet (lass 'fv')
for the funtion r -> K(r)
..............................................
Desription
r
distane argument r
theo
theoretial Poisson K(r)
border border-orreted estimate of K(r)
trans translation-orreted estimate of K(r)
iso
isotropi-orreted estimate of K(r)
..............................................
Default plot formula: .~r
where "." stands for 'iso', 'trans', 'border', 'theo'
Reommended range of argument r: [0, 0.25℄
Available range of argument r: [0, 0.25℄
An "fv" object is a data frame (that is, it also belongs to the class "data.frame") with attributes giving extra information such as the recommended way of plotting the function. One column of the data frame contains evenly spaced values of the distance argument r, while the other
columns contain estimates of the value of the function, or the theoretical value of the function under
CSR, corresponding to these distance values. The printout for KC above indicates that the columns
in the data frame are named r, theo, border, trans, and iso, and explains their contents. For
example, the column iso contains estimates of the K-function using the isotropic edge correction.
The function argument in an "fv" object is usually, but not always, called r.
7.5.2
Plotting "fv" objects
If f is an object of class "fv", then plot(f) is dispatched to the method plot.fv. The default
behaviour of plot(f) is to generate a plot containing several curves, each representing a different
edge-corrected estimate of the same target function, plotted against the distance argument r. For example, the command plot(Kest(swedishpines)) generates the left panel of Figure 7.19, which
shows three different estimates of the K-function for the swedishpines dataset, together with the
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Correlation
‘theoretical’ (expected) curve Kpois (r) = π r2 for CSR, all plotted against the distance argument r.
The legend indicates the meaning of each curve. The main title identifies the function object that
was plotted.
Kest(swedishpines)
1500
1500
K^ iso (r )
K^
(r )
trans
500
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1000
K^ iso (r )
1000
K pois (r )
0
0
500
K (r )
K^ b ord (r )
0
5
10
15
r (one unit = 0.1 metres)
20
0
5
10
15
20
r (one unit = 0.1 metres)
Figure 7.19. Plots produced by plot.fv. Left: default plot in response to the command
plot(Kest(swedishpines)). Right: plot of isotropic correction estimate Kbiso (r) against r.
The return value from plot.fv is a data frame containing more detailed information about the
meaning of the curves. For the left panel of Figure 7.19, the return value is
lty ol
label
meaning
1
1
itali(hat(K)[iso℄(r))
isotropi-orreted estimate of K(r)
2
1 itali(hat(K)[trans℄(r)) translation-orreted estimate of K(r)
3
1 itali(hat(K)[bord℄(r))
border-orreted estimate of K(r)
4
1
itali(K[pois℄(r))
theoretial Poisson K(r)
M
iso
trans
border
theo
SA
Here lty and ol are the graphics parameters controlling the line type and line colour, and label is
the mathematical notation for each edge-corrected estimate, in the syntax recognised by R graphics
functions.
The default plot in the left panel of Figure 7.19 can easily be modified. Set legend=FALSE to
suppress the legend, and main="" to suppress the main title. The legend position is automatically
computed to avoid overlap with the plotted curves, but this can be overridden by legendpos. For
further options see help(plot.fv).
The printout of an "fv" object indicates the range of values of r in the table as the ‘available
range’. It also gives a ‘reommended range’ which is generally shorter than the available range.
The default plot of the object will only show the function values over the recommended range and not
over the full range of values available. This is done so that the interesting detail is clearly visible in
the default plot. Values outside the recommended range may be unreliable due to increased variance
or bias, depending on the edge correction. To change the range of r values, use the argument xlim
in the plot command.
The variables plotted on the x and y axes are determined by the second argument to plot.fv,
which should be a formula, involving the names of columns of the object. The left side of the
formula represents what variables will be plotted on the y-axis, and the right side determines the
x variable for the plot. For example, in the object Kest(swedishpines), the column named iso
contains the values of the isotropic correction estimate. The command
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Spatial Point Patterns: Methodology and Applications with R
> Ks <- Kest(swedishpines)
> plot(Ks, iso ~ r)
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plots the isotropic correction estimate against r as shown in the right panel of Figure 7.19.
The many uses of formulae for controlling plots were introduced in Section 2.1.10. In plot.fv,
both sides of the plot formula are interpreted as mathematical expressions, so that operators like ‘+’,
‘-’, ‘*’, ‘/’ have their usual meaning in arithmetic. The right-hand side of the formula can be any
expression that, when evaluated, yields a numeric vector, and the left-hand side is any expression
that evaluates to a vector or matrix of compatible dimensions.
If evaluation of the left-hand side yields a matrix, then each column of that matrix is plotted
against the specified x variable as a separate curve. In particular the left-hand side of the formula
may invoke the function bind to indicate that several different curves should be plotted. For
example, to plot only the translation and isotropic correction estimators and the theoretical curve,
an appropriate syntax is: plot(Ks, bind(iso, trans, theo) ~ r) (results not shown).
The plot formula may also involve the names of constants like pi, standard functions like sqrt,
and some special abbreviations listed in Table 7.1. The symbol .x represents the function argument,
usually named r. One of the columns of function values will be designated as the ‘best’ estimate,
for use by some other commands in spatstat. This column is identified by the symbol .y. Some
or all of the columns of function values are designated as ‘acceptable’ estimates for the default plot,
and these are identified by the symbol ‘.’. The default plotting formula is ~ x indicating that
the acceptable estimates will be plotted against the function argument.
. .
argument of function
best estimate of function
all estimates of function for default plot
all columns of function values
upper and lower limits of shading
M
.x
.y
.
.a
.s
Table 7.1. Recognised abbreviations for columns of an "fv" object. To expand these abbreviations,
use fvnames(f, ".x") and so on.
SA
A plot formula can be used to specify a transformation that should be applied to the function
values before they are displayed. For example, to divide each of the function estimates by the
theoretical Poisson value, one could use plot(Ks, / theo ~ r). The resulting plot is shown
in the left panel of Figure 7.20. Alternatively one could plot the function estimates against the
Poisson value, using plot(Ks, ~ theo). This is shown in the right panel of Figure 7.20.
The mathematical labels for the plot axes, and for the individual curves, are constructed automatically by spatstat from the plot formula. If the plot formula involves the names of external
variables, these will be rendered in Greek where possible. For example, to plot the average number
of trees surrounding a typical tree in the Swedish Pines data,
.
.
> lambda <- intensity(swedishpines)
> plot(Ks, lambda * . ~ r)
Here we use the name lambda so that it will be rendered as the Greek letter λ in the graphics: the
y-axis will be labelled λ K(r).
b should be plotted against r2 .
In the discussion of Ripley’s paper, Cox [178] proposed that K(r)
This also linearises the plot of the K-function and has some theoretical support. This plot can easily
be generated, using the plot formula ~ r^2.
.
7.5.3
Manipulating "fv" objects
An "fv" object can be manipulated using the operations listed in Table 7.2.
223
0.8
1500
K^ iso (r )
K^
(r )
trans
K^ b ord (r )
K^ iso (r ) K pois (r )
K^
(r ) K (r )
trans
pois
1000
500
0.4
K (r )
0.6
K pois (r )
0.2
K (r ) K pois (r )
1.0
Correlation
K^ b ord (r ) K pois (r )
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0
0.0
K pois (r ) K pois (r )
0
5
10
15
20
r (one unit = 0.1 metres)
0
500
1000
1500
K pois (r )
Figure 7.20. Left: Estimates of K(r) divided by π r2 , plotted against r. Right: Estimates of K(r)
plotted against π r2 .
print a description
print a description
plot the function estimates
strip extra information (returns a data frame)
extract column named iso (returns a numeric vector)
extract subset (returns an "fv" object)
extract subset (returns an "fv" object)
perform calculations with columns of data frame
perform calculations with several "fv" objects
combine "fv" objects f1, f2, ...
combine an "fv" object f and data frame d
combine several redundant "fv" objects
check whether "fv" objects are compatible
make "fv" objects compatible
range of function values
apply smoothing to function values
derivative of function
compute Stieltjes integral with respect to f
convert to a function
SA
M
f
print(f)
plot(f)
as.data.frame(f)
f$iso
f[i,j℄
subset(f, ...)
with(f, expr)
eval.fv(expr)
bind(f1, f2, ...)
bind.fv(f, d)
ollapse.fv(f1, f2, ...)
ompatible(f1, f2, ...)
harmonise(f1, f2, ...)
min(f), max(f), range(f)
Smooth(f)
deriv(f)
stieltjes(g,f)
as.funtion(f)
Table 7.2. Operations for manipulating an "fv" object f.
Values of the function
An "fv" object is essentially a table, containing the values of the desired function (such as K(r)) at a
finely spaced grid of values of the function argument r. The function values can be extracted directly
from an "fv" object since it is also a data frame. A single column of values can be extracted using
the $ operator in the usual way: Ks$iso would extract a vector containing the isotropic correction
estimates of K(r).
The subset extraction operator ‘[’ has a method for "fv" objects. This always returns another
"fv" object, so it will refuse to remove the column containing values of the function argument r,
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Spatial Point Patterns: Methodology and Applications with R
for example. To override this refusal, convert the object to a data frame using as.data.frame and
then use ‘[’: the result will be a data frame or a vector.
Commands designed for data frames often work for "fv" objects as well. The functions head
and tail extract the top (first few rows) and bottom (last few rows) of a data frame. They also work
on "fv" objects: the result is a new "fv" object containing the function values for a short interval
of r values at the beginning or end of the range. The function subset selects designated subsets of
a data frame using an elegant syntax (see page 101) and this also works on "fv" objects. To restrict
Ks to the range r ≤ 0.1 and remove the border correction,
Convert to a true function
PL
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> Ko <- subset(Ks, r < 0.1, selet= -border)
The table of function values can also be converted to a true function in the R language using
as.funtion. This makes it easy to evaluate the function at any desired distance r.
> Ks <- Kest(swedishpines)
> K <- as.funtion(Ks)
> K(9)
[1℄ 129.096
By default, the result K is a function in R, with a single argument r (or whatever the original function
argument was called). The new function accepts numeric values or numeric vectors of distance
values, and returns the values of the ‘best’ estimate of the function, interpolated linearly between
entries in the table. If one of the other function estimates is required, use the argument value to
as.funtion to select it. Several estimates can be chosen:
M
> K <- as.funtion(Ks, value=".")
> K(9)
[1℄ 129.096
> K(9, "trans")
[1℄ 130.4998
Calculating with columns
SA
To manipulate or combine one or more columns of data in an "fv" object, it is typically easiest to
use with.fv, a method for the generic with. For example:
> Kr <- Kest(redwood)
> y <- with(Kr, iso - theo)
> x <- with(Kr, r)
The results x and y are numeric vectors, where x contains the values of the distance argument r,
and y contains the difference between the columns iso (isotropic correction estimate) and theo
(theoretical value for CSR) for the K-function estimate of the redwood seedlings data. For this to
work, we have to know that Kr contains columns named r, iso and theo. Printing the object will
reveal this information, as would typing names(Kr) or olnames(Kr).
The general syntax is with(X, expr) where X is an "fv" object and expr can be any expression involving the names of columns of X. The expression can include functions, so long as they are
capable of operating on numeric vectors. The expression can also involve the abbreviations listed in
Table 7.1. Thus: Ken <- with(Kr, . - pi*r^2) subtracts the ‘theoretical’ value from all the
available edge correction estimates. In this case the result Ken is an "fv" object. You can also get
a result which is a vector or single number:
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Correlation
> with(Kr, max(abs(iso-theo)))
[1℄ 0.04945199
Calculating with several "fv" objects
To manipulate or combine several "fv" objects, spatstat provides the special function eval.fv.
Its argument should be an expression involving the names of the "fv" objects which are to be
combined. For example, to find the difference between the K-functions of the redwood and ells
datasets,
PL
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> K1 <- Kest(redwood) ; K2 <- Kest(ells)
> DK <- eval.fv(K1-K2)
M
The result DK is another "fv" object. Note that something like eval.fv(Kest(redwood) Kest(ells)) would not work, because redwood and ells are not "fv" objects.
The objects in the expression should be ‘compatible’ in the sense that they have the same column
names, and the same vector of r values. However, eval.fv will attempt to reconcile incompatible
objects. (The spatstat generic function ompatible determines whether two or more objects are
compatible, and the generic function harmonise makes them compatible, if possible.)
Objects of class "fv" are data frames of function values, with extra attributes which explain the
function values and determine how they are plotted. The function eval.fv performs arithmetic on
the columns of function values, but also manipulates these extra attributes, so that when the result
of eval.fv is plotted, the labels on the plot are constructed in a meaningful way.
7.6 The pair correlation function
7.6.1
The pair correlation function of a point process
SA
A plot of the K-function can be difficult to interpret correctly in terms of the behaviour of the point
process, because of its ‘cumulative’ nature (Section 7.3.5). The value of K(r) contains contributions
for all interpoint distances less than or equal to r.
An alternative tool is the pair correlation function g(r) which contains contributions only from
interpoint distances equal to r. In two dimensions, it can be defined by
K ′ (r)
(7.22)
2π r
where K ′ (r) is the derivative of the K-function with respect to r. See [355, pp. 218–223, 232–
244]. The pair correlation function has a long history in astronomy [204, 317, 401, 552], mostly
for three-dimensional patterns. It has also arisen independently in geographical analysis and other
fields [285].
In geometric terms, K(r) is defined by drawing a circle of radius r centred on a point of the point
process, and counting the number of other points falling in the circle (see left panel of Figure 7.21).
To define g(r), draw two concentric circles of radius r and r + h, where h is a small increment of
distance, and count only those points falling in the ring between the two circles (see right panel
of Figure 7.21). This captures the interpoint distances di j that lie in the narrow range between
r and r + h. The expected count of such distances is λ K(r + h) − λ K(r). We standardise the
expected count by dividing it by the expected value for complete spatial randomness, which is
λ π (r + h)2 − λ π r2 , yielding
gh (r) =
g(r) =
λ K(r + h) − λ K(r) K(r + h) − K(r)
=
.
λ π (r + h)2 − λ π r2
2π rh + π h2
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Spatial Point Patterns: Methodology and Applications with R
Figure 7.21. Geometry of the K-function (Left) and the pair correlation function (Right).
SA
M
When h is very small, π h2 becomes negligible, and (K(r + h) − K(r))/h becomes the derivative of
K, so that gh (r) becomes the pair correlation function (7.22).
The concept of the pair correlation function has also been reinvented in different guises. The
O-ring statistic [700] is essentially K(r + h) − K(r), where now the ring thickness h is an algorithm
parameter that is held fixed and should not be too small. The O-ring statistic is approximately hg(r).
The K density [247] is defined as the derivative of the K-function, which is therefore exactly equal
to a multiple of the pair correlation function, K ′ (r) = 2π r g(r). Many writers seem to be unaware
of the existence of the pair correlation function and its long history in astronomy and other fields.
Imagine the observation window is divided into a fine grid of pixels. The probability that a given
.
.
pixel contains a random point is p = λ a where a is the pixel area. Here we use the symbol = to
mean that the quantities are equal when they are very small.
Now consider two pixels with centres u and v, separated by a distance r. See Figure 7.22. Let p2 (u, v) be
r
the probability that both pixels contain random points.
If the process is Poisson, then events in these pixels are
independent, and p2 (u, v) = p2 . For any stationary and
Figure 7.22. Interpretation of the pair isotropic point process, it turns out that
correlation g(r) as the (normalised)
probability that two pixels, a distance r
apart, both contain random points.
. p2 (u, v)
g(r) =
.
p2
(7.23)
That is, g(r) is the probability of observing a pair of
points of the process separated by a distance r, divided by the corresponding probability for a
Poisson process. In other words, if U and V are two ‘infinitesimal’ regions with areas du and dv,
respectively, separated by a distance r then
.
P {X has a point in U and a point in V } = λ 2 g(r) du dv
(7.24)
where λ is the intensity of the process.
The value g(r) = 1 is consistent with complete spatial randomness, because of the way the function has been standardised. A value g(r) < 1 indicates that interpoint distances equal to r are less
frequent than would be expected for a completely random process, so this suggests regularity. A
value g(r) > 1 indicates that this interpoint distance is more frequent than expected for a completely
random pattern, which suggests clustering. These interpretations are less ambiguous than the cor-
227
Correlation
responding statements for the K-function, because they refer only to interpoint distances equal to
r.
5
2.0
1
0.5
0.00 0.05 0.10 0.15 0.20 0.25
r
0
0.0
0.0
0.5
2
3
g (r )
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4
1.5
1.0
g (r )
1.0
g (r )
1.5
2.0
Notice that the value g(r) is not a correlation in the sense used by statisticians. The correlation
between two random variables is a number standardised to lie in the range between −1 and +1,
with the value 0 indicating a lack of correlation. It turns out to be impractical to standardise the
correlation structure of point processes in this way. Instead, g(r) is the kind of correlation often
used in physics: the possible values range from 0 to infinity, and the value 1 is associated with a
lack of correlation.
0.00 0.05 0.10 0.15 0.20 0.25
0.00 0.05 0.10 0.15 0.20 0.25
r
r
Figure 7.23. Empirical pair correlation function for each of the three patterns in Figure 7.1.
Figure 7.23 shows estimates of the pair correlation function for the three archetypal point patterns. Compare these with the empirical K-functions shown in Figure 7.7 on page 206.
SA
M
The left panel of Figure 7.7 shows that the K-function for the regular pattern is below the Poisson
curve for all distances r, suggesting (if we are not careful) that there is inhibition at all distances.
The plot of the pair correlation has a different message: g(r) is below the Poisson value, g(r) < 1,
for distances r up to 0.07 units, then it climbs steeply to a value greater than the Poisson value,
g(r) > 1 at about r = 0.10 units, before declining back to 1. In fact g(r) is zero for r ≤ 0.07 because
this extremely regular pattern has no interpoint distances shorter than 0.07 units. The reason for
the apparent discrepancy between g and K is clear: short interpoint distances are completely absent
in this pattern, and this deficit affects the cumulative count of pairwise distances in the K-function,
even at much larger distances r.
The K-function and the pair correlation function g are mathematically interrelated: g can be
obtained from K through (7.22), and K can be obtained from g by the reverse relationship
K(r) = 2π
Z r
sg(s) ds.
(7.25)
0
One may ask why anyone would use the K-function, if the pair correlation function g is easier to
interpret. The main reason is that K seems to perform better as a basis for statistical inference (such
as hypothesis tests).
Also, in theory g does not necessarily exist: it is essentially the derivative of the K-function, and
this does not exist if the K-function has jumps. For example, for a regular grid of points (randomly
translated to make it a stationary point process),√with points spaced at multiples of s units, the Kfunction has a jump at r = s and at r = 2s, r = s 2, and so on. This process has no pair correlation
function. This example could be realistic for some crystalline structures.
The practical interpretation of the pair correlation function is discussed in [355, sec. 4.3.4].
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Spatial Point Patterns: Methodology and Applications with R
7.6.2
Estimating the pair correlation function
A popular way to estimate the pair correlation function is by kernel smoothing [635, p. 126], [242,
257, 629, 637, 453, 531]. In very large samples, histogram-based methods could be used, and these
are common in astronomical applications.
Taking any edge-correction
estimator of the K-function of the general form (7.3), suppose we
replace the indicator 1 di j ≤ r by a kernel term κ (di j − r) to obtain a smooth estimate of K ′ (r),
then plug into the formula (7.22) for the pair correlation function, to obtain the ‘fixed-bandwidth’
kernel estimator [638, eq. (15.15), p. 284]
n n
|W |
∑ ∑κh (r − di j )ei j (r)
2π rn(n − 1) i=1
j=1
(7.26)
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gb(r) =
j6=i
where di j = kxi − x j k is the interpoint distance between the ith and jth data points, ei j (r) is an edge
correction weight, and κh is the smoothing kernel, with smoothing bandwidth h > 0.
The kernel κh is a rescaled version of a template kernel κ ,
1 x
κh (x) = κ
h
h
(7.27)
M
where κ is any chosen function that is a probability density over the real line with mean 0. For
example, κ could be the standard normal (Gaussian) density, so that κh would be the normal density
with mean 0 and standard deviation h. The convention followed in R and in spatstat is that the
template κ has standard deviation 1, so that κh has standard deviation h. The ‘smoothing bandwidth”
is the standard deviation of the smoothing kernel. (Note that other writers may use a different
convention.)
The usual choice of smoothing kernel for pair correlation functions is the Epanechnikov kernel
with half-width w,
3
x2
εw (x) =
(7.28)
1− 2
4w
w +
SA
where (x)+ = max(0, x). The kernel √
is a quadratic function truncated to the interval [−w, w]. The
standard deviation of εw (x) is h = w/ 5, so the kernel of bandwidth h is κh (x) = εh√5 (x).
To compute (7.26) in practice we must choose a bandwidth value h, and as is common in data
analysis, this choice involves a compromise between bias and variability. For excessively large
bandwidth h, variability will be well controlled but the shape of the function gb will be oversmoothed,
so that important detail in the pair correlation function may be missed. For excessively small bandwidth, the estimates gb(r) will have high variance, and the shape of gb will be erratic.
The standard rule
pof thumb [355, p. 236] is to take the half-width w of the Epanechnikov kernel
(7.28) to be w = c/ λ̂ where c is a constant between 0.1 and 0.2. This corresponds to taking the
bandwidth
c′
h= p
(7.29)
λ̂
√
where c′ = c/ 5 is between 0.045 and 0.090. Alternative bandwidth selection and bias correction
methods are discussed in [300], [355, p. 230 ff.]. The variance of gb(r) is discussed in [355, p. 232
ff.], [632]. The choice of denominator (related to estimation of the intensity) is also important to
controlling the variance, as mentioned in Section 7.4.6.
To compute the estimated pair correlation function in spatstat use pf.
> g <- pf(ells)
> plot(g)
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Correlation
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The function pf is generic, with a method for point patterns. By default, pf.ppp uses the estimator (7.26), with the Epanechnikov kernel, with bandwidth h selected by Stoyan’s rule of thumb
(7.29) with c = 0.15 so that c′ = 0.067.
In analysing a point pattern dataset, attention is often focused on the behaviour of g(r) for small
r, which reflects the correlation between close pairs of points. Unfortunately, estimation of g(r)
at small distances is intrinsically difficult, because geometry dictates that there will typically be
few observations at small distances. The estimator (7.26) has poor performance at small distances.
Theoretically, as r approaches zero, the variance of gb(r) becomes infinite, for many point processes
including CSR. The left panel of Figure 7.24 shows a real example where gb(r) has an infinite
asymptote at r = 0. These problems are due to the factor 1/r in (7.26). An alternative estimator
with usually better performance is
gb(r) =
1 n n κh (r − di j )
∑ ∑ di j ei j (r)
2π i=1
j=1
(7.30)
j6=i
10
8
6
5
10
0
0
5
10
15
r (km)
SA
r (km)
15
2
4
g (r )
0
M
0
2
4
g (r )
6
8
10
in which the contribution to gb(r) from an observed interpoint distance di j involves a factor 1/di j
rather than 1/r. This estimator can be invoked by specifying divisor="d" in pf.ppp. The right
panel of Figure 7.24 shows this estimator computed for the same real example.
Figure 7.24. Estimates of the pair correlation function of the Queensland copper deposits of Figure 1.11, using (Left) the divide-by-r estimator (7.26) and (Right) the divide-by-d estimator (7.30).
Another useful trick applies when an estimate of K(r) is already available: then we can calculate g(r) from K(r) through equation (7.22). This is particularly useful in large datasets, where
direct estimation of g(r) can be time-consuming. The conversion of K(r) to g(r) is done by the
method pf.fv, which computes the derivative by numerical differentiation using spline smoothing. Effectively the function K(r) is first smoothed by fitting polynomial approximations to K(r)
over successive ranges of r values. The derivatives of the polynomials are then computed, using
elementary calculus.
> K <- Kest(shapley)
> g <- pf(K, spar=0.5)
Since K is an object of class "fv", the last command is dispatched to pf.fv. Additional arguments
to pf.fv, such as the argument spar above, are passed to the function smooth.spline in the
stats package, which performs the spline smoothing. Using a little algebra, we may apply
psmoothing either to K(r) or to the transformed functions K(r)/(2π r), K(r)/(π r2 ), or L(r) = K(r)/π ,
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Spatial Point Patterns: Methodology and Applications with R
by specifying method="a", "b", "" or "d", respectively. Methods "b" to "d" offer improved
performance at small distances r. The default is method="" which seems to perform best overall.
See help(pf.fv) for more information.
7.7 Standard errors and confidence intervals
7.7.1
Variance under CSR
PL
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Statistical estimates should be accompanied by a measure of accuracy (such as a standard error) or
uncertainty (such as a confidence interval). This is not straightforward for the K-function: there is
b
no simple expression for the standard error or variance of K(r),
even in a Poisson process, because
b from different data points xi are not independent.
the contributions to K(r)
b assuming complete spatial randomness are available,
Approximations to the standard error of K(r)
using the techniques of U-statistics [576, 518, 433, 225]. Ripley [576] gave an approximate variance
estimate for the isotropic corrected K-function estimator,
biso (r) ≈ 2
var K
A
n−1
2 π r2 0.96Pr3
nPr5
+ 0.13 3
+
2
A
A
A
(7.31)
7.7.2
M
where n is the number of data points and A and P are the area and perimeter length of the window
W . Lotwick and Silverman [433] gave a more intricate and accurate approximation for the case of a
rectangular window. These approximations are chiefly useful for predicting the accuracy that can be
expected from a survey region of given size and shape. They are computed by Kest if the argument
var.approx=TRUE is given. For the L-function, the corresponding approximations for var b
Liso (r)
are calculated by the delta method.
Block bootstrap
SA
Bootstrap methods [250, 340] make it possible to estimate variance from the data, without assuming
the Poisson model. A very simple version of this approach is block variance estimation. Assuming
the window W is a rectangle, divide it into equal quadrats B1 , . . . , Bm . Suppose our estimator of K(r)
is (7.3). For a particular quadrat B, let
b B) = m|W | ∑ ∑ 1 di j ≤ r ei j (r)
K(r,
n(n − 1) xi ∈B j6=i
(7.32)
be an estimate of K(r) based only on the pairs (xi , x j ) for which xi falls in the quadrat B, while
b
b Bk ) over
x j may lie anywhere in W . The usual estimate K(r)
is the average of the estimates K(r,
all k = 1, . . . , m. We compute the pointwise sample mean, sample variance, and sample standard
deviation of these estimates, yielding an elementary bootstrap estimate [225, eq. (4.21), p. 52] of
b
b
the standard error for K(r).
Assuming K(r)
is approximately normally distributed, we can then
calculate pointwise 95% confidence intervals for the true value of K(r).
The block variance estimate is computed by the spatstat function varblok:
> swp <- resale(swedishpines)
> Kvb <- varblok(swp, Kest, nx=3, ny=3)
> plot(Kvb)
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Correlation
7.7.3
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The result is shown in the left panel of Figure 7.25. For a fixed value of r, the grey shading gives a
95% confidence interval for the true value of K(r). Formally this means that, in 95% of outcomes
of this procedure, the random interval computed here would embrace the true value of K(r).
b is
Notice that the confidence interval becomes wider as r increases, because the variance of K(r)
roughly proportional to r2 . If we use L(r) instead of K(r), the variance will be stabilised: var b
L(r) is
approximately constant as a function of r. The confidence intervals will then have roughly constant
width.
An important caveat is that these are pointwise calculations (i.e. performed separately for each
value of distance r) yielding pointwise confidence intervals (i.e. valid only for a single value of r at
a time). Our confidence that the true K-function lies entirely inside the grey shaded region, over all
values of r, is vastly less than 95%. A solution to this problem is explained below.
Confidence intervals from Loh’s bootstrap
A more flexible bootstrap approach was developed by Loh [430]. First we decompose the empirical
K-function (7.3) into the contributions from each individual data point xi ,
b xi ) = |W | ∑ 1 di j ≤ r ei j (r),
K(r,
n − 1 j6=i
for i = 1, . . . , n
(7.33)
b is the average of the estimates K(r,
b xi ) over all
called the local K-functions. The usual estimate K(r)
i = 1, . . . , n. We could again calculate the pointwise sample mean and sample variance of these local
K-functions. Instead, Loh’s approach is to resample from the suite of local K-functions. Choose an
independent random sample of size n, with replacement, from the numbers 1 to n. If the sample is
i1 , . . . , in , we calculate the average of the corresponding local K-functions,
1 n b
∑ K(r, xik ).
n k=1
M
K ∗ (r) =
(7.34)
b
This gives a resampled version of K(r).
Repeating this procedure a large number N of times, we
b
obtain a bootstrap sample of the distribution of K(r).
Pointwise confidence intervals can then be
computed directly from the observed quantiles of the bootstrap sample. This calculation is performed by the spatstat function lohboot:
SA
> Kloh <- lohboot(swp, Kest)
> plot(Kloh)
The result is shown in the right panel of Figure 7.25. The default behaviour is to compute a pointwise
95% confidence interval.
The grey-shaded intervals in Figure 7.25 are pointwise confidence intervals. Ideally we would
prefer a global confidence interval or confidence band, having a 95% chance of containing the
entire graph
p of the true K-function. This is possible in Loh’s approach. First we replace K(r) by
L(r) = K(r)/π to stabilise the variance. For each resampled L-function L∗ (r), we calculate the
deviation
D∗ = max |L∗ (r) − b
L(r)|,
0≤r≤rmax
b over the interval of r values
the maximum vertical separation between the graphs of L∗ (r) and L(r)
up to a nominated maximum distance rmax . If there are N resampled L-functions, this gives N
deviation values; we take the 95th percentile of these deviations, say D0.95 , and construct the global
confidence band with boundaries
L− (r) = b
L(r) − D0.95
b + D0.95 .
L+ (r) = L(r)
(7.35)
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0
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5
10
K (r )
10
5
K (r )
15
Spatial Point Patterns: Methodology and Applications with R
0.0
0.5
1.0
1.5
2.0
r (metres)
0.0
0.5
1.0
1.5
2.0
r (metres)
Figure 7.25. Pointwise 95% confidence intervals for the true K-function of the Swedish Pines using
(Left) the spatial block bootstrap varblok, and (Right) Loh’s bootstrap lohboot. Solid lines:
estimated K-function. Dashed lines: the theoretical K-function under CSR, i.e., K(r) = π r2 .
This has the desired interpretation that, in 95% of outcomes of this procedure, the true L-function
will lie entirely between the two limits L− (r), L+ (r). This procedure is performed by lohboot with
the argument global=TRUE:
M
> Lg <- lohboot(swp, Lest, global=TRUE)
A global confidence band for K(r) can then be obtained by back-calculation:
> Kg <- eval.fv(pi * Lg^2)
0.5
1.0
5
0
0.0
0.0
10
K (r )
1.0
0.5
L (r )
1.5
15
2.0
SA
2.5
The results are plotted in Figure 7.26.
1.5
r (metres)
2.0
0.0
0.5
1.0
1.5
2.0
r (metres)
Figure 7.26. Global 95% confidence interval for the true L-function (Left) and back-transformed
confidence interval for the K-function (Right) of the Swedish Pines, using Loh’s bootstrap.
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Correlation
7.8 Testing whether a pattern is completely random
7.8.1
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b
The K-function estimated from a point pattern dataset, K(r),
is often compared graphically with
the theoretical K-function for the homogeneous Poisson process, Kpois (r) = π r2 , serving as the
benchmark of a completely random pattern. In the examples above, large discrepancies between
b
K(r)
and Kpois (r) suggested that the Swedish Pines are not completely random. However, even
b and Kpois ,
with a completely random pattern, we will never obtain perfect agreement between K
b and
because of random variability. We need a rule for deciding whether the difference between K
Kpois is large enough (relative to the scale of variability) to convince us that the point process is not
b and Kpois is statistically significant.
completely random: that is, whether the discrepancy between K
This rule is a significance test or hypothesis test. The principles of statistical tests are explained in
Chapter 10; here we explain the basic idea and show how to conduct the test in spatstat.
Pointwise envelopes
0.5
1.0
r (metres)
1.5
2.0
0
2
4
6
8
10 12
0.0
K (r )
M
8
6
SA
0
2
4
K (r )
10 12
In order to assess the statistical significance of deviations between the observed K-function and
the theoretical K-function for CSR, essentially we need to know how much variability should be
b if the pattern is completely random.
expected in the estimate K(r)
0.0
0.5
1.0
1.5
2.0
r (metres)
Figure 7.27. Construction of envelopes of the K-function. Left: the K-function estimate of the
Swedish Pines dataset (thick line), and of 39 simulated realisations of CSR with the same number
of points (thin lines). Right: grey shading between the upper and lower envelopes (maximum and
minimum values) of the K-functions of the simulated patterns.
The left panel of Figure 7.27 shows the K-function estimated from the rescale swedishpines
dataset (thick line), and also the K-functions of 39 simulated realisations of CSR with the same
number of points (thin lines). Each thin line could have been generated by
plot(Kest(runifpoint(npoints(swp), Window(swp))))
b
The spread of the thin lines typifies the variability in K(r)
that we should expect if the pattern is
completely random with the same number of points as the data. Clearly, the estimate of K(r) for the
Swedish Pines data lies outside this range, for some values of r.
In the right panel of Figure 7.27 we have shaded the region between the highest and lowest lines,
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Spatial Point Patterns: Methodology and Applications with R
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that is, the minimum and maximum values of the K-functions of the simulated patterns, called the
upper and lower envelopes of the simulated functions.
The envelope plot can be interpreted as a statistical significance test, as follows. Without peeking
at the data, choose a particular distance r, say r∗ = 1 metre. If the swedishpines data were
completely random, then the data and simulated patterns would be statistically equivalent, since
they would all be completely random patterns with the same number of points. Consequently the
b ∗ ) for the data and simulated patterns would be statistically equivalent numbers. By
values of K(r
symmetry, since there are 40 values altogether (one data value and 39 simulated values), there would
be a chance of 1 in 40 that the value for the data is the smallest, i.e. that the data value is smaller
than all of the simulated values. Similarly there would be 1 chance in 40 that the data value is larger
than all the simulated values. This makes 2 chances in 40, or a probability of 1/20 = 0.05 that the
data value lies outside the range of the simulated values purely by chance. This has in fact occurred
in Figure 7.27, so we could declare the result to be statistically significant with a p-value of 0.05.
This is an example of a Monte Carlo test, a test based on simulations from the null hypothesis,
using a symmetry principle. A full explanation of Monte Carlo tests is given in Chapter 10.
The spatstat function envelope computes simulation envelopes. The right panel of Figure 7.27 was generated by the commands
> E <- envelope(swp, Kest, nsim=39, fix.n=TRUE)
> plot(E)
The result of envelope is an object of class "envelope" which also belongs to the class "fv". It
can be plotted, printed, and manipulated using the tools for "fv" objects described in Section 7.5.
See Table 7.2 on page 223. The print method gives a lot of detail:
> E
SA
M
Pointwise ritial envelopes for K(r)
and observed value for 'swp'
Edge orretion: "iso"
Obtained from 39 simulations of CSR
Alternative: two.sided
Signifiane level of pointwise Monte Carlo test: 2/40 = 0.05
.......................................................
Desription
r
distane argument r
obs observed value of K(r) for data pattern
theo theoretial value of K(r) for CSR
lo
lower pointwise envelope of K(r) from simulations
hi
upper pointwise envelope of K(r) from simulations
.......................................................
Default plot formula: .~r
where "." stands for 'obs', 'theo', 'hi', 'lo'
Columns 'lo' and 'hi' will be plotted as shading (by default)
Reommended range of argument r: [0, 2.4℄
Available range of argument r: [0, 2.4℄
Unit of length: 1 metre
Arguments to envelope allow us to specify the data pattern, the summary function to be used, the
number of simulations, one-sided or two-sided tests, and so on. Instead of testing CSR, we can use
envelope to test essentially any null hypothesis. Full details of the envelope function are given
in Section 10.8. Table 10.1 on page 397 lists the most useful arguments to envelope.
Envelopes computed in the manner of Figure 7.27 are called pointwise envelopes because the
maximum and minimum values are calculated separately for each distance r (i.e. for each ‘point’
on the horizontal axis). A very important caveat about pointwise envelopes is that they can only
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Correlation
be interpreted as a statistical significance test if the distance value r∗ was chosen in advance. It is
not valid to first generate the Figure and then choose a favourable value of r; this would be ‘data
snooping’. If we allow ourselves this freedom then we are much more likely to find a value of r
where the empirical K(r) lies outside the envelopes, so the result is much less significant. The same
caveat is emphasised in [572, 431, 40] and is discussed at length in Chapter 10.
7.8.2
Global envelopes
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The problem of ‘data snooping’ can be avoided using global envelopes, shown in Figure 7.28. As
distinct from the pointwise envelopes in Figure 7.27, the global envelopes in Figure 7.28 delimit a
zone of constant width. The width is determined by finding the most extreme deviation from the
theoretical K-function that is achieved by any of the simulated K-functions, at any distance r along
the horizontal axis.
That is, for each simulated dataset, we compute the maximum vertical deviation D between the
b and Kpois over some range of distances. The maximum Dmax of the deviations for all
graphs of K
simulated datasets is taken. The global envelopes are
E− (r) = Kpois (r) − Dmax
E+ (r) = Kpois (r) + Dmax .
(7.36)
If the graph of the estimated K-function for the data transgresses these limits, it is statistically
significant with a p-value of 1/(m + 1) where m is the number of simulated patterns. Taking m = 19
gives a test with significance level 0.05.
The left panel of Figure 7.28 shows global envelopes for the K-function computed by
> E <- envelope(swp, Kest, nsim=19, rank=1, global=TRUE)
0.0
0.5
1.0
1.5
r (metres)
2.0
0.0 0.5 1.0 1.5 2.0 2.5
L (r )
10
0
5
K (r )
SA
15
20
M
A more powerful test is obtained if we (approximately) stabilise the variance, by using the Lfunction in place of K, as shown in the right panel of Figure 7.28. This clearly rejects the null
hypothesis of CSR at the 0.05 significance level. The two panels used the same set of simulated
point patterns.
0.0
0.5
1.0
1.5
2.0
r (metres)
Figure 7.28. Global envelopes relating to the Swedish Pines data. Left: global envelopes for the
K-function. Right: global envelopes for the L-function. The same simulated patterns were used for
both panels.
Global envelopes were proposed by Ripley over 30 years ago [572, 574, 575], but they are still
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Spatial Point Patterns: Methodology and Applications with R
not widely used in applications, despite the fact that they provide a simple solution to the data
snooping problem.
7.8.3
Non-graphical tests
Statistical tests can also be carried out without needing any graphical display. Two useful procedures
are the MAD (maximum absolute deviation) test, and the DCLF (Diggle-Cressie-Loosmore-Ford)
test, described in Sections 10.7.3 and 10.7.4, respectively. The MAD test [575] is equivalent to
plotting the global envelopes described above and declaring a statistically significant outcome if the
estimated K-function (etc.) wanders outside the global envelope anywhere:
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> mad.test(swp, Lest, nsim=99, rmax=2, use.theo=TRUE)
Maximum absolute deviation test of CSR
Monte Carlo test based on 99 simulations
Summary funtion: L(r)
Referene funtion: theoretial
Alternative: two.sided
Interval of distane values: [0, 2℄ metres
data: swp
mad = 0.29999, rank = 1, p-value = 0.01
M
The DCLF test [223, p. 122], [190, eq. (8.5.42), p. 667], [431] is based on the mean squared deviation between the empirical summary function and the theoretical function over a range of distance
values. The DCLF test is more powerful than the MAD test provided the maximum distance is
chosen carefully; the MAD test is more reliable when little information is available about the range
of dependence [40].
> dlf.test(swp, Lest, nsim=99, rmax=2, use.theo=TRUE)$p.value
[1℄ 0.01
SA
Both tests indicate there is strong evidence that the Swedish Pines data are not completely random.
7.9 Detecting anisotropy
A point process is ‘isotropic’ if all its statistical properties are unchanged when it is rotated (Section 5.6.3). If there is any statistical property of the process that does change when the process is
rotated, then the process is ‘anisotropic’. A point process could be stationary, but not isotropic.
The microstructure of a mineral could be homogeneous, but with a preferential direction, perhaps
reflecting the history of deformation [519, 355].
The following code generates a synthetic point pattern which is homogeneous and anisotropic.
First we generate a Simple Sequential Inhibition process, and then deform space by squashing the
y-axis.
> X <- rSSI(0.05, win=owin((0,1), (0, 3)))
> Y <- affine(X, mat=diag((1, 1/3)))
Figure 7.29 shows the simulated pattern Y and its Fry plot, clearly indicating that the pattern is
anisotropic.
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Correlation
Figure 7.29. Anisotropy. Left: synthetic point pattern exhibiting anisotropy. Right: Fry plot.
7.9.1
Sector K-function
SA
M
Various modifications of the K-function have been suggested [520, 636] for measuring anisotropy.
Instead of counting all data points falling inside
a circle of radius r centred at a data point, we could
replace the circle by another geometrical shape.
The sector shown in Figure 7.30 is that part of
the disc of radius r lying between two lines at orientations α and β . Here r is the function argument,
while the orientations α and β are fixed. The sector
K-function of a stationary point process is 1/λ times
the expected number of points lying within this sector, given that the vertex of the sector is a point of the
Figure 7.30. Geometry of the sector Kprocess. The sector K-function can be estimated using
function.
the same edge-correction techniques as the original Kfunction.
The spatstat function Ksetor computes the empirical sector K-function using the translation
edge correction, together with the theoretical value for a Poisson process, Ksector, pois (r) = (β −
α )r2 /2 if α , β are expressed in radians.
For example, to compute the sector K-functions for two 30-degree angle sectors centred on the
x and y axes, respectively:
> Khoriz <- Ksetor(Y, begin = -15, end = 15, units="degrees")
> Kvert <- Ksetor(Y, begin = 90-15, end = 90+15, units="degrees")
Note that angles are measured anticlockwise from the x-axis. Anisotropy would be suggested if
these two functions appeared to be unequal. We may then compare them by superimposing the
plots, or by using eval.fv to compute the difference between the functions. The left panel of
Figure 7.31 shows the result of
> plot(Khoriz, trans/theo ~ r, lty=2)
> plot(Kvert, trans/theo ~ r, add=TRUE)
where we have divided each estimate by the theoretical value.
The right panel of Figure 7.31 shows the difference Khoriz-Kvert and a 95% confidence interval computed by the following code:
238
Spatial Point Patterns: Methodology and Applications with R
horizontal − vertical
0.8
0.4
−1
−2
−3
−4
−5
0.0
K sector (r )
0
0.10
r
0.20
0.00
0.02
0.04
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0.00
r
Figure 7.31. Anisotropy analysis using Ksetor for the synthetic pattern in Figure 7.29. Left:
Superimposed plot of normalised Ksetor functions for 30-degree sectors centred on the horizontal
axis (dashed line) and vertical axis (solid line). Right: 95% confidence interval for the difference
(×104 ) between horizontal and vertical sector K-functions using block bootstrap.
M
> dK <- funtion(X, ...) {
K1 <- Ksetor(X, ..., begin = -15, end = 15, units="degrees")
K2 <- Ksetor(X, ..., begin = 90-15, end = 90+15, units="degrees")
eval.fv(K1-K2)
}
> CIdK <- varblok(Y, dK, nx=5)
The sector K-function requires a choice of the limiting angles α and β , so is best used when there
is some prior knowledge about the preferential directions in the pattern.
7.9.2
Pair orientation distribution
SA
Anisotropy can also be measured by observing using
the directions of arrows joining pairs of points.
Consider all pairs of points in a point pattern that
lie more than r1 and less than r2 units apart, where
r1 < r2 are fixed distances. For each such pair, we
measure the direction of the arrow joining the points,
as an angle in degrees anticlockwise from the x-axis.
The probability distribution of these angles is the point
pair orientation distribution.
Figure 7.32 shows the geometry for estimating the
Figure 7.32. Geometry for the point pair
cumulative
distribution function of the angles, called
orientation distribution function.
the ‘point pair orientation distribution function’ [638,
(14.53), p. 271]
E number of points in O(u, r1 , r2 , φ ) X has a point at location u
,
Or1 ,r2 (φ ) =
E number of points in O(u, r1 , r2 , 360) X has a point at location u
0 ≤ φ ≤ 360
(7.37)
where O(u, r1 , r2 , φ ) is the region bounded by circles of radius r1 and r2 centred at u and lines
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Correlation
at orientation 0 and φ through u, shown in Figure 7.32. As Figure 7.32 suggests, this summary
function can be estimated in the same way as the K-function.
The spatstat function pairorient computes an estimate of either the cumulative distribution
function or the probability density of angles. The default is to compute the probability density, using
a kernel smoothing estimate:
> f <- pairorient(Y, r1=0.02, r2=0.05, sigma=5)
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The result is an "fv" object. Although this could be plotted by plot.fv, the display is challenging
to interpret. For probability distributions of angles, it is usually better to display a rose diagram or
rose of directions plot, in which the function value is interpreted as radial distance from the centre
of the plot.
120
105
90
75
60
135
150
165
45
30
15
180
0
195
345
210
330
225
315
255
270
285
300
M
240
Figure 7.33. Rose diagram of estimated probability density for the point pair orientation distribution of the synthetic pattern in Figure 7.29.
SA
Figure 7.33 shows a rose diagram for the estimated probability density f, generated by rose(f).
It strongly indicates a preferential direction around 90 and 270 degrees. That is, at distances between
0.02 and 0.05, pairs of data points tend to be either directly above or directly below each other, and
are rarely found directly to one side of each other.
The point pair orientation distribution can be an effective tool for finding a preferential direction,
but seems to be quite sensitive to the choice of the distances r1 and r2 . An alternative tool is the
nearest-neighbour orientation distribution (Section 8.8).
7.9.3
Anisotropic pair correlation function
These considerations lead us back to the Fry plot (Section 7.2.2). In a stationary point process,
virtually all information about the correlation structure is contained in the Fry plot. For example,
the K-function could be calculated from the Fry plot, by counting how many dots in the Fry plot
fall in a circle of radius r centred at the origin, and standardising appropriately. Replacing the circle
by another ‘test set’, such as a sector or a ring, gives the other summary functions described in this
chapter.
By counting dots in the Fry plot we are effectively treating the Fry plot as a point process in its
own right, and finding the intensity of this process. The intensity measure of the Fry plot, appropriately standardised, is called the reduced second moment measure of the original point process. It
generalises Ripley’s K-function from circles to test sets of general shape [674, 520, 638].
Definition 7.2. For a stationary point process X with intensity λ , the reduced second moment
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Spatial Point Patterns: Methodology and Applications with R
measure is the measure defined for any test set A in two-dimensional space by
K (A) =
1 E number of points of X (except u) falling in A + u X has a point at u
λ
(7.38)
for an arbitrary location u.
1.05
1
0.6
0.95
0.9
SA
0.5
0.7
M
0.8
0.9
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For example, if we take the test set A = b(0, r), the disc of radius r centred at the origin, then
A + u is the disc of radius r centred at u, and equation (7.38) reduces to (7.4), so that K (A) = K(r),
the value of Ripley’s K-function for distance r. See Section 7.13 for details.
The measure K can be regarded as the generalisation of the K-function, and also as a standardised version of the intensity measure of the Fry plot. The connection between (7.38) and the Fry
plot is that a point x j in X falls in A + u if and only if the vector difference x j − u falls in A.
Be warned that some writers use the term reduced second moment measure when they mean the
K-function. This has caused confusion. As originally defined [457, 197], the reduced second moment measure is a measure, while the K-function is a function obtained by evaluating this measure
for discs of increasing radius.
In most applications we can assume that the Fry plot has an intensity function, so that K has
an intensity or density function, the anisotropic pair correlation function g(v) defined for vectors v.
This can be estimated by kernel smoothing of the Fry plot with edge correction and standardisation.
The left panel of Figure 7.34 shows an estimate of the anisotropic pair correlation function for the
point pattern data in Figure 7.29.
Figure 7.34. Anisotropic pair correlation function. Left: image of estimated anisotropic pair correlation g(u) for the synthetic pattern in Figure 7.29, as a function of vector difference u, with origin
at centre of image. Detail of function inside the square of side length 0.2 centred at the origin.
Right: anisotropy ratio r(u) = g(u)/g(kuk) where g is the rotational mean of g.
The anisotropic pair correlation function g(v) is a generalisation of the pair correlation function
g(r) to anisotropic, stationary point processes. It allows the correlation between pairs of points to
be dependent on their relative position (the vector v) and not only on the distance r between them.
Again, g(v) is standardised so that it is equal to 1 when the process is completely random.
The anisotropic pair correlation function can be interpreted as a joint probability, as in Figure 7.22 on page 226. Imagine the observation window is divided into a fine grid of pixels, and
consider two pixels u and v separated by the vector v − u. If p2 (u, v) is the probability that both
pixels contain random points, then
. p2 (u, v)
g(v − u) =
(7.39)
p2
where p2 is the corresponding probability for a Poisson process.
241
Correlation
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The spatstat package provides a function Kmeasure which computes an estimate of the
anisotropic pair correlation function (the density of the reduced second moment measure K ), effectively by smoothing the Fry plot. The algorithm approximates the point pattern and its window
by pixel images, introduces a Gaussian smoothing kernel, and uses the Fast Fourier Transform to
compute a density estimate of K , effectively using the translation edge correction. The standard
deviation σ of the kernel must be chosen by hand; there do not seem to be any established rules for
bandwidth selection in this context.
The left panel of Figure 7.34 shows the estimated anisotropic pair correlation function g(v) as
a function of the two-dimensional vector v, with the origin in the centre of the picture. This is
effectively a smoothed and renormalised version of the Fry plot. It was generated by the commands
> ganiso <- Kmeasure(Y, sigma=0.02, eps=0.001)
> detail <- square((-0.1,0.1))
> plot(ganiso[detail℄)
Choosing the bandwidth sigma=0.02 and pixel resolution eps=0.001 ensures that we do not oversmooth or pixellate interesting detail, which (according to the Fry plot) is at a scale of about 0.1
units.
Evidence for anisotropy can be judged by comparing the value of g(v) for different vectors v of
the same length but different orientations. A useful tool is to compute the rotational mean
g(t) =
1
2π
Z 2π
g((t cos θ ,t sin θ )) dθ
(7.40)
0
M
which gives, for each distance t, the average value of g(v) over all vectors v of length kvk = t. Then
the anisotropy ratio r(v) = g(v)/g(kvk) is a comparison of the anisotropic pair correlation with its
rotational mean.
The spatstat function rotmean computes the rotational mean of any image. The right panel
of Figure 7.34 shows the result of plotting the anisotropy ratio g(v)/g(kvk), computed by
> giso <- rotmean(ganiso, result="im")
> grel <- ganiso/giso
> plot(grel[detail℄)
SA
The right panel of Figure 7.34 gives a clear impression of anisotropy, with anisotropy ratios above 1
in the vertical direction and below 1 in the horizontal direction, at distances of about 0.04 units. The
left panel shows that this corresponds to pair correlation values closer to 1 in the vertical direction,
and further away from 1 in the horizontal direction, at this distance range. This is consistent with
the appearance of the Fry plot.
For a stationary isotropic point process there is a close relationship between the K-function and
the isotropic pair correlation function through equations (7.22) and (7.25). Equation (7.25) can be
generalised to a stationary, anisotropic point process:
K (A) =
In particular, for A = b(0, r),
K(r) =
Z
Z
g(v) dv.
(7.41)
A
g(v) dv.
(7.42)
b(0,r)
Equation (7.41) can be used to compute an estimate of the second moment measure K (A) for any
region A. We first use Kmeasure to compute an estimate of g as a pixel image, then compute the
integral of the pixel image over the domain A using integral.im. For example, if A is the square
window detail created above,
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Spatial Point Patterns: Methodology and Applications with R
> integral(ganiso, detail)
[1℄ 0.03681638
The pair correlation function also has a direct relationship to the second moment of the number
of points falling in a region B. If N = n(X ∩ B) then N(N − 1) is the number of ordered pairs of
distinct points falling in B; the expected number of such pairs is
E[N(N − 1)] = λ 2
Z Z
B B
g(v − u) du dv
(7.43)

E
∑
∑
xi ∈X x j ∈X
x j 6=xi
PL
E
as we can see intuitively by dividing B into a fine grid of pixels, and summing over all pairs of
distinct pixels the joint probability that both pixels contain random points.
The result above is a special case of Campbell’s formula for second moments. Consider the sum,
over all pairs of distinct random points xi , x j where i 6= j, of some function f (xi , x j ). The second
moment version of Campbell’s formula states that


Z Z

2
f (xi , x j )
=
λ
f (u, v)g(v − u) du dv

(7.44)
M
provided the expectation is finite. This is the analogue of equation (6.11) on page 169, for second
moments.
7.10 Adjusting for inhomogeneity
If a point pattern is known or suspected to be spatially inhomogeneous, then our statistical analysis
of the pattern should take account of this inhomogeneity. For general discussion see [355, sec. 4.10],
[230, 225].
The general pair correlation function
SA
7.10.1
First we need to extend the concept of the pair correlation function to point processes which cannot
be assumed to be stationary or isotropic.
Suppose the point process X has intensity function λ (u), so that
En(X ∩ A) =
Z
λ (u) du
A
for any bounded region A. Additionally suppose that X has second moment intensity function
λ2 (u, v), also called the product density, defined as the function that satisfies
E[n(X ∩ A) n(X ∩ B)] =
Z Z
A B
λ2 (u, v) du dv
(7.45)
for any disjoint regions A and B. For a Poisson process with intensity function λ (u), we have
λ2 (u, v) = λ (u)λ (v).
The second moment intensity function has a simple interpretation. Again imagine that the observation window is divided into a fine grid of pixels of area a. Let p(u) be the probability that the pixel
.
with centre u contains a random point. Then p(u) = λ (u)a. Let p2 (u, v) be the joint probability that
.
the two pixels with centres u and v both contain random points. Then p2 (u, v) = λ2 (u, v)a2 .
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Correlation
In this general setting we can define the general pair correlation function
g2 (u, v) =
λ2 (u, v)
λ (u)λ (v)
(7.46)
for any two different spatial locations u and v. Using the approximations above, g2 (u, v) is approximately p2 (u, v)/(p(u)p(v)), the probability that pixels centred at u and v both contain random
points, divided by the corresponding probability for a Poisson process with the same intensity function λ (u).
Inhomogeneous K and g functions
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7.10.2
Estimation of the function g2 (u, v) is not practically possible without some further assumption on
the form of the function. A strategy proposed in [46] effectively assumes that the point process is
correlation-stationary4
g2 (u, v) = g(v − u),
(7.47)
that is, the pair correlation between u and v depends only on their relative position. This would be
true if the process is stationary, but is also true for an inhomogeneous Poisson process, and for many
other processes [678].
Assuming (7.47), it is possible to define a counterpart of K(r) called the inhomogeneous Kfunction. The idea is that each point xi will be weighted by wi = 1/λ (xi ), the reciprocal of the
intensity at xi , and each pair of points xi , x j will be weighted by wi j = wi w j = 1/(λ (xi )λ (x j )).
The inhomogeneous K-function is defined as
"
#
1
Kinhom (r) = E ∑
(7.48)
1 0 < ||u − x j || ≤ r u ∈ X
x ∈X λ (x j )
M
j
SA
assuming this does not depend on location u. If (7.47) holds, then (7.48) does not depend on u.
Thus, Kinhom (r) is the expected total ‘weight’ of all random points within a distance r of the
point u, where the ‘weight’ of a point xi is 1/λ (xi ). If the process is stationary, then λ (u) is constant
and Kinhom (r) reduces to the usual K-function (7.6).
For an inhomogeneous Poisson process with intensity function λ (u), the inhomogeneous Kfunction is
(7.49)
Kinhom, pois (r) = π r2
exactly as for the homogeneous case.
The standard estimators of K can be extended to the inhomogeneous K-function:
1 ||xi − x j || ≤ r
1
b
Kinhom (r) = p
∑ bλ (x )bλ (x ) e(xi, x j ; r)
D |W | ∑
i j6=i
i
j
(7.50)
where e(u, v, r) is an edge correction weight as before, and b
λ (u) is an estimate of the intensity
function λ (u). The constant D p in (7.50) is the pth power of
D=
1
1
∑
|W | i b
λ (xi )
(7.51)
which has expected value 1 if the intensity is estimated without error. Choosing the power p equal to
1 or 2 improves statistical performance. Theoretical results in [46] show that, if the intensity funcbinhom (r) is an unbiased estimator
tion is known, or can be estimated with very high accuracy, then K
of Kinhom (r) when p = 0, and is approximately unbiased when p = 1 or 2.
4 Called
‘second order intensity-reweighted stationary’ (soirs) in [46].
244
Spatial Point Patterns: Methodology and Applications with R
1
M
1.5
2
2.5
3
PL
E
In practice, the intensity function will need to be estimated from data, so the estimator (7.50)
could be biased. The intensity function is usually estimated from the same point pattern data,
which can lead to substantial bias in the estimate of Kinhom (r). This justifies the data-dependent
denominator D. Nonparametric estimates of intensity, being more responsive to local fluctuations
in the data, tend to produce more biased estimates of Kinhom (r) than parametric estimates of intensity,
provided the intensity model is correctly specified [46, 225, 274].
The estimator (7.50) requires the estimated intensities b
λ (xi ) at the data points xi . To reduce bias
it is advisable to use a leave-one-out intensity estimator as described in Section 6.5.1.3.
The empirical inhomogeneous K-function is computed in spatstat by the command Kinhom(X,
lambda) where X is the point pattern and lambda is the estimated intensity function. Here lambda
may be a pixel image, a funtion(x,y) in the R language, a fitted point process model, a numeric
λ (xi ) at the data points xi only, or it may be omitted (and will then be esvector giving the values b
timated from X). By default, the data-dependent denominator D1 is used: the power p is controlled
by the argument normpower.
Figure 7.35. Full dataset of Japanese Black Pines of Numata and Ogata (Left) and kernel-smoothed
intensity estimate (Right). Smoothing bandwidth selected by likelihood cross-validation.
SA
Figure 7.35 shows the full dataset of Japanese Black Pine seedlings and saplings in a 10 metre square study region recorded by Numata [503] and extensively studied by Ogata [514, 516].
A kernel-smoothed intensity estimate is shown, with smoothing bandwidth selected by likelihood
cross-validation using bw.ppl (see Section 6.5.1.2 on page 170):
> numata <- residualspaper$Fig1
> lambda <- density(numata, bw.ppl)
The chosen bandwidth seems appropriate, and the intensity estimate gives a strong impression of
inhomogeneity. We now compute the inhomogeneous K-function:
> numataK <- Kinhom(numata, lambda)
The result is shown in Figure 7.36. The edge-corrected estimates are close to the Poisson theoretical
value, suggesting that the data are consistent with an inhomogeneous Poisson process with the
intensity shown in Figure 7.35.
If the intensity function is not given in the call to Kinhom then it will be estimated by kernel
smoothing using the leave-one-out estimator. We could have obtained the same result by
> numataK <- Kinhom(numata, sigma=bw.ppl)
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Correlation
20
The argument sigma=bw.ppl is passed to density.ppp. In practice the results obtained from this
procedure can be quite sensitive to the choice of bandwidth.
trans
K^ inh om(r )
10
b ord
K^ inh om(r )
b ord m
K^ inh om (r )
0
0.0
PL
E
pois
K inh
om(r )
5
K inh om (r )
15
iso
K^ inh om(r )
0.5
1.0
1.5
2.0
2.5
r (metres)
Figure 7.36. Inhomogeneous K-function for full Japanese Pines data, using the estimated intensity
function from the right panel of Figure 7.35.
The inhomogeneous analogue of the L-function is
p
Linhom (r) = Kinhom (r)/π .
(7.52)
M
For an inhomogeneous Poisson process, Linhom, pois (r) ≡ r. The corresponding empirical L-function
is again justified by the fact that it approximately stabilises the variance. The inhomogeneous Lfunction can be computed in spatstat using Linhom, which has the same arguments as Kinhom.
The relationship (7.42) between the K-function and pair correlation function of a stationary
point process extends to the inhomogeneous K-function:
SA
Kinhom (r) =
Z
b(0,r)
g2 (0, v) dv.
(7.53)
In general g2 (0, v) depends on the orientation of v as well as the distance kvk. The rotational mean
of the pair correlation
Z
1 2π
g(r) =
g2 (0, (r cos θ , r sin θ )) dθ
2π 0
is the ‘inhomogeneous pair correlation function’ ginhom (r), and satisfies
ginhom (r) =
K ′ inhom (r)
.
2π r
The inhomogeneous pair correlation function is computed by pfinhom:
> g <- pfinhom(bei)
The previously mentioned method pf.fv, which converts a K-function estimate into a pair correlation function estimate by numerical differentiation, also works for estimates of the inhomogeneous
K-function. Thus the following is an alternative to the foregoing estimation procedure:
> g <- pf(Kinhom(bei))
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Spatial Point Patterns: Methodology and Applications with R
To construct confidence intervals for the true value of Kinhom (r) or ginhom (r) one can use varblok
or lohboot as described in Section 7.7.
To test the hypothesis that the point process is an inhomogeneous Poisson process, use envelope,
dlf.test, or mad.test. Some care is required: as explained in Chapter 10, it is important to ensure that the simulated patterns are treated in exactly the same way as the original data pattern was
treated. Figure 7.37 shows the global envelopes of the centred inhomogeneous L function for the
full Japanese Pines data of Figure 7.35, generated by
PL
E
> lam <- density(numata, bw.ppl)
> E <- envelope(numata, Linhom, sigma=bw.ppl,
simulate=expression(rpoispp(lam)),
use.theory=TRUE, nsim=19, global=TRUE)
> plot(E, . - r ~ r)
0.05 0.10 0.15
−0.15
M
−0.05
L inh om (r ) − r
The plot indicates significant evidence against the inhomogeneous Poisson model, with a suggestion
that there is inhibition at small distances.
0.0
0.5
1.0
1.5
2.0
2.5
r (metres)
SA
Figure 7.37. Centred inhomogeneous L-function Linhom (r)−r for the Numata-Ogata Japanese pines
data (black line) and global envelopes from 19 simulated realisations of an inhomogeneous Poisson
process with intensity estimated by a leave-one-out kernel smoother.
We emphasise that — despite its name — the inhomogeneous K-function does not apply to
every spatially inhomogeneous point process; it applies only when the point process is correlationstationary, equation (7.47). Substantial differences between the estimates of the inhomogeneous
K-function obtained from different subsets of the data, or obtained using different edge corrections,
would suggest that this assumption is false. A formal hypothesis test for correlation-stationarity is
described in Section 16.8.5.
7.10.3
Local scaling
The inhomogeneous K-function effectively assumes that the spatial scale of interaction remains
constant, while the intensity is spatially varying. This is not an appropriate assumption for the
bronze filter data in Figure 1.9 on page 8, an inhomogeneous point pattern in which the spacing
between points increases gradually from left to right.
An alternative approach to inhomogeneity [312, 556, 309, 311] is to assume that the point process is equivalent, in small regions, to a rescaled version of a ‘template’ process, where the template
247
Correlation
process is stationary and isotropic, and the rescaling factor can vary from place to place. This could
be an appropriate model for Figure 1.9.
We would then be assuming that, for two locations u and v sufficiently close together,
ku − vk
g(u, v) = g1
(7.54)
s
PL
E
where g1 is the pair correlation function of the ‘template’ process, and s is the local scaling factor
applicable to both locations u and v. Rescaling the spatial
√ coordinates by a factor 1/s rescales
the intensity by s2 , so the appropriate value is s = 1/ λ where λ is the local intensity in the
neighbourhood of u and v.
In practice, we would first estimate the intensity function of the original data by b
λ (u), then for
each pair of data points xi , x j define the rescaled distance
di∗j =
where the rescaling factor is
kxi − x j k
s(xi , x j )


1
1 1
.
q
+q
s(xi , x j ) =
2
b
b
λ (x j )
λ (xi )
M
An edge-corrected estimator of Ripley’s original K-function is then applied to the distances di∗j to
give the locally scaled K-function [312, 556, 311].
The spatstat package provides the commands Ksaled and Lsaled which compute the
locally scaled K- and L-functions. Their syntax is similar to Kinhom.
To estimate a locally scaled K-function for the bronze filter data (Figure 1.9), we first estimated
the intensity, assuming it is an exponential function of the x-coordinate, by fitting a point process
model (see Chapter 9):
> X <- unmark(bronzefilter)
> fit <- ppm(X ~ x)
> lam <- predit(fit)
SA
The locally scaled K-function was then estimated by
> Kbro <- Ksaled(X, lam)
The result is plotted in Figure 7.38.
Again we emphasise that the locally scaled K-function is applicable only when the point process
is locally scaled. A formal hypothesis test for local scaling is described in Section 16.8.5.
7.11 Local indicators of spatial association
Another tool for handling inhomogeneous patterns is a LISA (Local Indicator of Spatial Association), obtained by decomposing a summary statistic into contributions from each of the data points.
b xi ) from each
For example, the empirical K-function (7.3) can be broken into contributions K(r,
b
individual data point xi , shown in (7.33), known as the local K-functions. The usual estimate K(r)
b xi ) over all i = 1, . . . , n.
is the average of the estimates K(r,
If it is suspected that the pattern may be a patchwork of different textures in different places, or
248
8
6
4
0
2
K scaled (r )
10
Spatial Point Patterns: Methodology and Applications with R
0.5
1.0
1.5
2.0
PL
E
0.0
r (normalised units)
Figure 7.38. Locally scaled K-function of the bronze filter data.
M
that the pattern contains some anomalous features, then it can be useful to classify the n local Kb x1 ), . . . , K(r,
b xn ) into several groups of functions, perhaps using hierarhical clustering
functions K(r,
techniques. This approach was proposed independently in [281, 20] and developed in [186, 185].
Repeated measures analysis or functional principal component analysis [562] could be used to identify the main differences between the local functions.
The spatstat functions loalK, loalL, loalpf compute local versions of the K-function,
L-function, and pair correlation function, respectively. Of course the L-function is not a sum of
contributions from individual data points: the local L-function is arbitrarily defined as b
L(r, xi ) =
b xi )/π )1/2 . There are also inhomogeneous counterparts loalKinhom, loalLinhom, and
(K(r,
loalpfinhom which are the contributions to the inhomogeneous K-function, etc, from each data
point.
Local K-functions of the Swedish Pines data are computed by
> lK <- loalK(swedishpines)
SA
The result is an "fv" object with n + 2 = 73 columns, where n = 71 is the number of data points.
b xi ) for each data point, in order. The final two
The first n columns give the local K-functions K(r,
columns contain the distance values r and the K-function for a Poisson process. Extracting only the
function values by
> loK <- as.data.frame(lK)[, fvnames(lK, ".a")℄
> rr
<- with(lK, r)
gives the data in a format ready for cluster analysis or functional data analysis. Normally loK
would be transposed to t(loK) so that each row of the matrix contains one local K-function. Hierarchical clustering of the local K-functions, based on the mean square distance between functions,
is implemented by
> loH <- hlust(dist(t(loK)))
A plot of loH would show a dendrogram of the local K-functions. See [186, 185] for further
discussion.
The average of the local K-functions for the data points falling in a region A can also be computed by Kest using the argument domain.
249
Correlation
7.12 Third- and higher-order summary statistics
T (r) =
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We saw on page 211 that two different point processes can have identical first-order (intensity) and
second-order (correlation) properties. In order to distinguish between such processes we need another approach. One possible strategy is to estimate the third-order moments, which would involve
counting triples rather than pairs of data points.
For a stationary point process X with intensity λ , the triangle-counting function T (r) is defined
as the normalised expected number of triangles formed by points of X, with all side lengths less
than or equal to r, with one corner at a specified point of the process:

n

n
1 

E  ∑ ∑m(xi , x j , u) | u ∈ X
λ3
i=1 j=1
(7.55)
j6=i
where m(a, b, c) is the maximum side length of the triangle with vertices a, b, c,
m(a, b, c) = max{ka − bk, ka − ck, kb − ck }.
For a homogeneous Poisson process with intensity λ we have
(7.56)
M
√
3 3 4
π
)r .
Tpois (r) = (π −
2
4
0.008
0.004
0.000
T (r )
SA
Formal theory and edge corrections are given in [600].
Estimates of the triangle-counting function T (r) are computed in spatstat by Tstat. For the
point pattern Xell shown in Figure 7.11, the empirical estimate Tb(r) computed by Tstat(Xell)
is shown in Figure 7.39. This shows clear deviation from the values expected for a Poisson process.
Simulation envelopes can be computed in the usual way.
0.00
0.10
0.20
r
Figure 7.39. Triangle-counting function Tb(r) for the simulated realisation of the cell process in
Figure 7.11. Solid line: translation correction estimate. Dashed line: theoretical value for Poisson
process.
Three-point correlation functions are frequently used in astronomy [537] and even n-point correlation functions are used [378, sec. 3.2], [445], [355, pp. 4.4.2].
250
Spatial Point Patterns: Methodology and Applications with R
7.13 Theory∗
This section covers some basic theory for second moments of point processes. For more detail,
see [484, Appendix C], [355, sec. 1.5], [27, 32, 569], [576, chap. 3]. For a full account, see [198,
199].
7.13.1
Second moment measures
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Let X be a point process. We are interested in the variance of the count n(X ∩ B),
var n(X ∩ B) = E[n(X ∩ B)2 ] − [En(X ∩ B)]2
and the covariance of two such counts,
cov[n(X ∩ B1 ), n(X ∩ B2 )] = E[n(X ∩ B1 )n(X ∩ B2 )] − [En(X ∩ B1 )] [En(X ∩ B2 )].
A key observation is that n(X ∩ B1 )n(X ∩ B2 ) is equal to the number of ordered pairs (x, x′ ) of points
in the process X such that x ∈ B1 and x′ ∈ B2 .
Definition 7.3. Let X be a point process in R2 . Then X × X is a point process on R2 × R2 consisting
of all ordered pairs (x, x′ ) of points x, x′ ∈ X. The intensity measure ν2 of X × X is a measure on
R2 × R2 satisfying
ν2 (A × B) = E[NX (A)NX (B)].
M
This measure ν2 is called the second moment measure of X.
The second moment measure contains all information about the variances and covariances of
the variables NX (A). Campbell’s formula (6.11) applied to X × X becomes
E[ ∑
∑
f (xi , y j )] =
xi ∈X x j ∈X
Z
Z
R2 R2
f (u, v) dν2 (u, v)
SA
for a measurable function f : R2 × R2 → R.
For example, for the uniform Poisson point process of intensity λ > 0 in Rd , the second moment
measure satisfies ν2 (A × B) = λ 2 |A| |B| + λ |A ∩ B|. To simplify the calculation of certain moments,
we introduce the second factorial moment measure
ν[2] (A × B) = E[n(X ∩ A)n(X ∩ B)] − E[n(X ∩ A ∩ B)].
This is the intensity measure of the process X ∗ X of all ordered pairs of distinct points of X. It
satisfies
Z Z
E[ ∑ ∑ f (xi , x j )] =
f (u, v) dν[2] (u, v).
xi ∈X x j ∈X, j6=i
R2 R2
The name ‘factorial’ is derived from
ν[2] (A × A) = E[n(X ∩ A)2 ] − E[n(X ∩ A)] = E[n(X ∩ A)[n(X ∩ A) − 1]].
For example, for the uniform Poisson process of intensity λ , the second factorial moment measure
is ν[2] (A × B) = λ 2 |A||B|.
∗ Starred sections contain advanced material, and can be skipped by most readers.
251
Correlation
Definition 7.4. A point process X on R2 is said to have second moment intensity λ2 if
ν[2] (C) =
Z
C
λ2 (u, v) du dv
(7.57)
for any compact C ⊂ R2 × R2 .
Informally, λ2 (u, v) gives the joint probability that there will be points of X in infinitesimal
regions around two specified locations u and v:
.
P {N( du) > 0, N( dv) > 0} = λ2 (u, v) du dv.
PL
E
For example, the uniform Poisson process has second moment density λ2 (x, y) = λ 2 . The binomial
process of n points in W has λ2 (u, v) = n(n − 1)/|W |2 if x, y ∈ W , and zero otherwise.
Definition 7.5. Suppose X is a point process on R2 which has an intensity function λ (x) and a
second moment intensity λ2 (u, v). Then we define the pair correlation function of X by
g2 (u, v) =
λ2 (u, v)
.
λ (u)λ (v)
For a uniform Poisson process of intensity λ , we have λ (u) ≡ λ and λ2 (u, v) ≡ λ 2 , so that
g2 (u, v) ≡ 1. For a binomial process of n points in a region W , we have g2 (u, v) ≡ 1 − 1/n.
7.13.2
Second moments for stationary processes
For a stationary point process in R2 , there is a ‘disintegration’ of the second moment measure.
M
Theorem 7.1. Let X be a stationary point process on R2 with intensity λ . Then there is a measure
K on R2 such that, for a general integrand f ,
"
#
Z Z
E
∑ ∑
f (xi , x j ) = λ
f (u, u + v) dK (v) du.
(7.58)
xi ∈X x j ∈X, j6=i
K is called the reduced second moment measure of X.
SA
To understand the measure K , we notice that for A, B ⊂ R2
"
Z Z
λ |A|K (B) =
1A (u)1B (v − u) dν[2] (u, v) = E
∑
∑
xi ∈X x j ∈X, j6=i
#
1A (u)1B (v − u) .
This may also be obtained directly from (7.58) by taking f (u, v) = 1A (u)1B (v − u). Since λ |A| =
En(X ∩ A), we have
E ∑xi ∈X∩A n(X ∩ (B + xi ) \ xi )
K (B) =
.
(7.59)
En(X ∩ A)
The right-hand side of (7.59) may be interpreted as the average, over all points xi of the process, of
the number of other points x j of the process such that x j − xi ∈ B.
For the uniform Poisson process, K (B) = λ |B|.
Suppose X is a stationary process on R2 which has a second moment density function λ2 . Then
by comparing (7.57) with (7.58) we can see that λ2 (u, v) depends only on v − u, say λ2 (u, v) =
λ 2 g(v − u) for some function g, and we can write
K (B) = λ
Z
g(u) du.
B
This is the fundamental relationship between the pair correlation and the second moment measure.
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Spatial Point Patterns: Methodology and Applications with R
7.14 FAQ
• Should I use Ripley’s K-function, the inhomogeneous K-function, or the scaled K-function?
This depends on the assumptions that are appropriate for your data. Using the K-function assumes
the point process has homogeneous intensity and is correlation-stationary. Using the inhomogeneous K-function assumes the process is correlation-stationary. Hypothesis tests for checking
these assumptions are described in Section 16.8.5.
PL
E
• When I plot the estimated K-function of my data using the command plot(Kest(X)), what is the
quantity on the horizontal axis, and what units is it expressed in?
The horizontal axis is distance between points. If the plot annotation does not indicate the units
in which the distance is measured, then the point pattern dataset X did not include a specification
of the units of length. This should be repaired using the function unitname.
• When I plot the estimated K-function of my data using the command plot(Kest(X)), the horizontal axis is the distance in metres, but what is the quantity on the vertical axis, and what units
is it expressed in?
The quantity on the vertical axis is the average number of neighbours of a typical point, divided by
the intensity (average number of points per unit area) so that different patterns can be compared.
It is measured in units of area (number of points divided by points-per-unit-area). The reference
benchmark is that for a completely random process the value of K(r) is the area of the disc of
radius r.
M
• When I plot the estimated K-function of my data using the command plot(Kest(X)), the scale
marks on the y-axis are huge numbers like 1e9. Is this wrong? I don’t see anything like this in
your book.
SA
The values of K(r) are areas, and should be of the same order as the area of the observation window, expressed in the units you are using for the spatial coordinates. If your window is 30 kilometres across and the coordinates are recorded in metres, the window area is about 300002 ≈ 109
square metres. To avoid numerical overflow, it would be wise to rescale the spatial coordinates,
for example converting metres to kilometres.
• When I plot the estimated K-function of my data using the command plot(Kest(X)), I don’t
understand the meaning of the different coloured curves.
These curves are different estimates of the K-function, using different ‘edge correction’ techniques. Usually one of the curves is the theoretical value of the K-function, K(r) = π r2 , corresponding to a completely random pattern.
The accompanying legend (plotted unless legend=FALSE) gives a mathematical expression for
each of the edge corrections, and shows the corresponding line colour and line type used to plot
the estimate.
The return value of plot.fv is a table giving the line colour, line type, keyword, mathematical
expression, and long text description of each curve that was plotted. For further information about
the different edge corrections, see help(Kest).
The estimates of K(r) by different edge correction techniques should be roughly equal. If the
curves for the isotropic correction (iso), translation correction (trans), and border correction
(border) estimates are wildly different, this suggests that estimation of K is difficult for these
data (e.g., because there are too few data points, or the window is very narrow, or there are data
points very close to the edge of the window).
253
Correlation
For information on how to modify the plot of the K-functions, see help(plot.fv) or the examples in help(Kest).
• I have computed the inhomogeneous K-function using Kinhom. The different edge-corrected estimates have very different values. Which correction should I use?
PL
E
Discrepancies between the edge-corrected estimates of the inhomogeneous K-function computed
by Kinhom(X, lambda) usually occur when the intensity estimate lambda is inaccurate, or when
the data are not correlation-stationary. First check that the estimated intensity appears to be reasonably accurate. If so, try changing the arguments normalise and normpower. If large discrepancies still remain, check for data points very close to the window boundary, which are treated
differently by the different edge corrections. If there are no such points, the tentative conclusion
is that the assumption of a correlation-stationary process is violated. Consider performing a test
of this assumption (Section 16.8.5).
• I can’t seem to control the range of r values in plot(Kest(X)). How can I control it? How is
the default plotting range determined?
Use the argument xlim to control the range of the x-axis, as documented in help(plot.fv). For
example, plot(Kest(X), xlim=(0, 7)).
An object of class "fv" contains function values for a certain range of r values (the ‘available range’, which is usually the maximum possible range). However, the default behaviour
of plot.fv is to plot the function values for a narrower range of r values (the ‘recommended
range’) which usually contains the important detail in the function. Both the available range and
recommended range are printed when the "fv" object is printed.
M
Using the argument xlim when plotting the K-function will override the recommended range.
However, obviously we cannot choose xlim to be wider than the available range of r values. To
extend the available range, you would need to re-compute the K-function, specifying the argument
r. This should be a vector of closely spaced values of r ranging from 0 to the desired maximum.
• When plotting a summary function using plot.fv, how can I control the legend position, text
size, and other parameters?
See the help file for plot.fv. To set the legend text to half normal size, for example, set
legendargs=list(ex=0.5).
SA
To move the legend outside the plot area, call plot.fv with legend=FALSE, save the result, and
use the columns in this data frame in a subsequent call to the legend function.
• I have plotted a summary function using a formula such as sqrt(./pi)
save the plotted curves as another summary function?
~ r. Is it possible to
Yes, if the right-hand side of the formula is the function argument. Use with.fv, for example
with(Kest(ells), sqrt(./pi))
• I used the envelope command to compute envelopes of the L-function. The printout for the
envelope object says that it contains ‘pointwise envelopes’. What value of r was used for these?
‘Pointwise’ means that for each chosen value of the distance r, the range of simulated values of
b has been calculated, and a test based on these values would be valid. Plotting the pointwise
L(r)
envelopes shows what the result of the test would have been if we had chosen any particular value
of r. For further explanation, see Chapter 10.
• Is it possible to combine the empirical K-functions from several point patterns into a common
K-function?
Yes; see Section 16.8.1.
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Spatial Point Patterns: Methodology and Applications with R
• I have mapped the locations of bird nests on a small island. Do I need to use an edge correction
for K(r)?
SA
M
PL
E
This is an example of the ‘Small World Model’ (page 145). Strictly speaking the K-function is
not defined because the point process is not stationary. A sensible strategy is to use the empirical
K-function without edge correction (Kest with orretion="none") and to base inference on
simulations from the appropriate null hypothesis.